I think it's really normal. My son, at almost 6, got his first set of stitches a couple weeks ago... he's had his share of bumps and bruises, but hasn't had anything too terrible happen until recently. Even so, I have had nightmares about something terrible happening to him since he was about 2 years old. He's an active, strong, persistent, learn- the-hard-way kind of kid and took a while to develop fear of anything.
I didn't let it influence how I parented him too much though. I was careful to make sure I could always sprint to him before he reached the street when he would run off chasing crows, and didn't let him get in places on the big playground where I couldn't get to him quickly or get him down if I needed to. I've let him tear down grass hills on his bike even though it almost makes my heart stop. We do have rules about certain things though... like if he wants to climb a tree he has to be able to get up there himself or he isn't big enough to do it yet; and if he can get up there he can get down (I am still spotting him in case of a fall).
I guess it comes down to knowing your kid and knowing what they are able to do, and trusting that because of that they'll almost always be ok. You have to let them do and try and explore or you wind up with a child that is too afraid to try anything.
Take a breath Momma~ You're normal :)