You can use the word "but" as a transitional word, but it is recommended to not use contractions. Likewise, your sentence will make more sense by changing doesn't to does not.
The sentence we are completing is formed by two clauses with contrasting ideas. Thus, we should choose a transitional word that helps convey contrast. The best ones would be "however" and "but". We also need to pay attention to punctuation. If we were to use "however", we would need to change the punctuation. Therefore, to maintain the sentence the way it is, the only possible option is "but".
She tries to be patient but doesn't always succeed.
Churchill asks a rhetorical question so that he can then express what he sees as the goal that must be shared by all free nations: the security, safety, and freedom of everyone in the entire world.
