"Conflict" means struggling or fighting. The word "conflict" sounds clean and neutral. It's the word that a psychologist might use to talk about fighting with people, for example.
If you "shy away" from something, it means that you try to avoid it. For example, if a friend of yours often says controversial things, you can say this about him:
He's never been one to shy away from controversy. That's for sure!
When you "shy away from" conflict, it meant that you try to avoid arguing or disagreeing with people directly. The phrase "shy away from" makes this sound a little weak and cowardly.