I think the amount you fall and the severity of it depends a lot on what horse you're riding, and how experienced you are with handling spooks/bucks/rears/etc.
Someone could have ridden a perfect horse for 10 years and not really be able to stay on for the thrills because they've never had to. Or, you could have someone who's ridden for a couple years but has been riding more spirited horses and has learned to stay on better. Of course, sometimes it's near impossible to stay on because it took you by surprise, or your horse has decided to be a rodeo star, or any number of reasons.
I also feel like horse riding is inherently more dangerous than a lot of other sports because you're relying on a prey animal that thinks on its own.
When I first started riding my mare, I probably fell off more in two months than I had in my whole six years of riding beforehand. And that was because she was much more spirited than any horse I had ever been on so I lacked the knowledge and muscles to stay on when she spooked or bucked or refused a jump.
Since I've started riding her, I've learned how to stay on much better. Sometimes it's because I'm aware that she could do something (spook at something, refuse a jump), so I'm prepared enough to expect it. Other times, I've just gotten more leg strength and have been able to think quickly when something does happen. For example, a month or so ago, my horse decided to let out a huge buck, bigger than she's probably ever done before. Because I'm used to her antics, I was able to stay in the saddle and keep riding like it didn't happen.
The last time I fell off was because I was attempting a lead change in a small arena, and when I put my weight in the saddle, the saddle started slipping, spooking her, and since I was sideways (and I am not a trick rider) I fell off. The last time I fell off because of something she did (which was still my fault) was back in March and she refused a jump. I've only really fallen twice this year! Not including me being an idiot bareback and kinda just sliding off... twice.
All that being said, I've never actually hurt myself from falling (knock on wood!). Never broken a bone, had stitches, or anything more than a cut or bruise (and lots of soreness). I've never had to go to the doctor because of a fall. I have no idea why that is since I'm one of the least coordinated people on this planet, but I am not complaining.
Your first fall I think is always scary. Try not to worry too much about falling. It will probably happen again, but don't worry about it until it's happening or else you'll not enjoy riding as much. Have fun, relax, and don't do anything too crazy. Just wear a helmet, always. Also, triple check your girth before you get on