Constant battling inside yourself is confusing, frustrating, exhausting, and makes some people wonder if they are crazy.
I think you answered your own question really well. It is probably true that both of these sides of your are a part of who you are, but maybe you keep them seperate as a way to deal with things? i don't know. You would have a btter idea than anyone.
As for it being normal or not? Hey what's normal? LOL No such thing as far as I am concerned, but I do know what you are asking.
The research I have done on this subject myself, and what I have learned in college working toward my degree, that this is not that uncommon. The reasons it happens can vary though and of course each person is unique.
You wanted to know if you should see a doctor, therapist, or something like that? I would say that it can't hurt. If you can find someone (like a counselor/therapist) that you like and feel you can build trust with them, they may be able to help you resolve some of what you are struggling with.
If these inner struggles really bother you and effect your life, you might want help with them. Not because you are crazy or "not normal" but just because we all need help sometimes with things we can't handle alone.
I hope this helps.
Take care of yourself!