Take another look at the sentence:
With sharp scissors, Thinh tried to even her client's hair, but the woman kept twisting in her seat to yell at her two small children who were terrorizing the salon, Thinh finally decided that crooked ends were better than poking out a client's eye.
Thinh tried to even her client's hair is a main clause. The woman kept twisting in her seat begins another main clause. You can correctly connect two main clauses with a comma and a coordinating conjunction like but.