Yeah, that's definitely normal. One thing you can do to combat it is to get some fingerboard lubricant... Fast Fret works good. This lessens the friction that the strings have on your fingers allowing them to get smoother without tearing as much. It's also good for your strings because it keeps them clean and keeps them from oxidizing. Also, it makes your neck play really smooth. There are different kinds.... some come in spray cans some come in stick applications. Just check some out though, I think you'll like the way it feels.