How do you write “conquer” in Korean?

I think 극복 will be the best word.

정복/정복하다 also means conquest/conquer, but is used more for conquests of war, conquering other countries, etc. 극복 is used for overcoming difficulties, fears, etc.

Actually, 극복하다 is the verb and 극복 is the noun, but standing alone I think that the simpler noun form 극복 is much more appropriate. (If you wanted to use a verb you'd want a different form like the imperative 극복하라).