WikiRhymer has about 700 rhyming words and phrases for "thirst" starting with...

burst
    cursed
      durst
        first
          nursed
            pursed
              versed
                worst
                  coerced
                    conversed
                      disbursed
                        dispersed
                          emersed
                            headfirst
                              immersed
                                rehearsed
                                  reversed
                                    submersed
                                      traversed
                                        interspersed
                                          reimbursed
                                            unrehearsed
                                              Hearst
                                                Hirst
                                                  Hurst
                                                    Kerst
                                                      Kirst
                                                        Wurst
                                                          • at first
                                                          • first things first
                                                          • do one's worst
                                                          • at the worst
                                                          • if worst comes to worst
                                                          Pure Rhyme Set #1 of 1 (Add to/Edit Set Members)
                                                          Words that have a pure rhyme on their last syllable only.
                                                          Whitehurst
                                                            Whitehurst's
                                                              End Rhyme Set #1 of 3 (Add to/Edit Set Members)
                                                              Dewhirst
                                                                Dewhurst
                                                                  End Rhyme Set #2 of 3 (Add to/Edit Set Members)
                                                                  the hurst
                                                                    outburst
                                                                      shoot first
                                                                        bratwurst
                                                                          fred durst
                                                                            head first
                                                                              traversed
                                                                                man first
                                                                                  cloudburst
                                                                                    twenty first
                                                                                      Bathurst
                                                                                        Lyndhurst
                                                                                          Bringhurst
                                                                                            Amherst
                                                                                              Elmhurst
                                                                                                Allenhurst
                                                                                                  End Rhyme Set #3 of 3 * (Add to/Edit Set Members)

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                                                                                                  Words that "almost" rhyme on the vowel-based rhyme sound of the stressed syllable like: be/eat or maybe/shapely. (Click for more info)
                                                                                                  burse
                                                                                                    curse
                                                                                                      firsts
                                                                                                        hearse
                                                                                                          merce
                                                                                                            nurse
                                                                                                              perce
                                                                                                                pers
                                                                                                                  perse
                                                                                                                    purse
                                                                                                                      terse
                                                                                                                        verse
                                                                                                                          worse
                                                                                                                            adverse
                                                                                                                              adverse
                                                                                                                                averse
                                                                                                                                  coerce
                                                                                                                                    converse
                                                                                                                                      disburse
                                                                                                                                        disperse
                                                                                                                                          diverse
                                                                                                                                            emerse
                                                                                                                                              immerse
                                                                                                                                                inverse
                                                                                                                                                  obverse
                                                                                                                                                    perverse
                                                                                                                                                      rehearse
                                                                                                                                                        reverse
                                                                                                                                                          submerse
                                                                                                                                                            transverse
                                                                                                                                                              traverse
                                                                                                                                                                intersperse
                                                                                                                                                                  reimburse
                                                                                                                                                                    Bearce
                                                                                                                                                                      Pearse
                                                                                                                                                                        • visiting nurse
                                                                                                                                                                        • line one's purse
                                                                                                                                                                        • in reverse
                                                                                                                                                                        • from bad to worse
                                                                                                                                                                        • go from bad to worse
                                                                                                                                                                        • for the worse
                                                                                                                                                                        • for better or for worse
                                                                                                                                                                        • take a turn for the worse
                                                                                                                                                                        • turn for the worse
                                                                                                                                                                        • all the worse
                                                                                                                                                                        Near Rhyme Set #1 of 43 (Add to/Edit Set Members)
                                                                                                                                                                        bursts
                                                                                                                                                                          firsts
                                                                                                                                                                            Hearst's
                                                                                                                                                                              Near Rhyme Set #2 of 43 (Add to/Edit Set Members)
                                                                                                                                                                              berths
                                                                                                                                                                                births
                                                                                                                                                                                  earth's
                                                                                                                                                                                    earths
                                                                                                                                                                                      worths
                                                                                                                                                                                        rebirths
                                                                                                                                                                                          Worth's
                                                                                                                                                                                            Near Rhyme Set #3 of 43 (Add to/Edit Set Members)

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                                                                                                                                                                                            Rhymes made up of more than one word. For instance, "jealous" and "tell us" or "shaky" and "make me."

                                                                                                                                                                                            No mosaic rhymes for one-syllable words.