I'll weigh in with my personal experience. From about 6-11 I had some mild desires to do "girlish" things and some curiosity about girl's clothing. At 11(ish) when the big T got flowing during puberty, I started experimenting with crossdressing but even then it wasn't so much a gender thing. In my adult life while I knew that my crossdressing was more than just a fetish, I still classified it as such. As I look back on it now, I was experiencing dysphoria but denied it, repressed it and in some cases didn't recognize it as such. It wasn't until last year in my very late 30's that I finally recognized and admitted that I'm trans.
As previously mentioned, from what I've learned there are a lot of people who don't realize the dysphoria until their middle-aged or even older. Not everyone has the "I'm a person of Gender A stuck in the body of Gender B" experience in their youth. I've talked directly with many trans-women who didn't even think about their gender until they were well into adulthood. So from that my conclusion is yes you're totally normal.