This is great example of a subjective topic where there really isn't a wrong or right answer. Who is to define what normal is? Who is to say what entities remain in the dimension of normal?
What I can tell you, is by the society I live in if there were a vote put up in which the majority dictates if your situation is normal or not, they would likely vote that it isn't normal to watch anime. I live in southern U.S. and most of the people where I live believe video games are still only for children.
My opinion though? I think they are weird shows. Does that make the people who watch it weird or "not normal"? Maybe. Maybe not, just depends on the viewer. But the act of watching anime itself in my eyes is weird, simply because its not something the people around me in my neck of the woods don't do.