To Conquer
する verb
克服 is "conquest" so the verb version of that is to conquer.
こくふくする
This is a jukugo word that uses the on'yomi readings of the kanji. You should be able to read this on your own.
三歳の頃犬に噛まれてから、ずっと犬に恐怖心を抱いていたんですが、この小さな仔犬に出会って犬嫌いを克服しました。
I’ve been afraid of dogs since I was bitten by one when I was three, but I’ve overcome my hatred for them since meeting this little puppy.