If you face your demons, you confront your fears or something that you have been trying hard to avoid.
[From UsingEnglish.com]
Demon itself is described as "a negative feeling that causes you to worry or behave badly":
She had her demons and, later in life, they drove her to drink.
[From Cambridge Dictionaries Online]
The phrase can vary: fight one's (own) demons, overcome one's (own) demons or deal with one's (own) demons are some options.
