R.J.
I agree with everything Susan said, except 1 thing, I rode both western and english, and for *me* english was FAR easier. :D :D :D I suspect it's personal preference.
I started riding draft horses at age 5 (now there's a sight, we're talking 18 hands high), did western for about a year at age 8, went back to English (dressage & jumping) at 9, exercised racehorses in Del Mar at 12 & 13, and went cross country at 14-18 (I moved around a lot... I rode whatever was available). Polo I didn't start till my 30's.
You fall off. You get stepped on (huzzah for boots, even rubber ones are better than sneakers). You have close calls with fingers. There's an old saying that 'you're not a horseman till you've fallen off 7 times'. But kids don't fall, as a rule, because they can't. Not unless they try. Kids in lessons are put on SEDATE horses. The calmest, gentlest, most boring horses on the planet. And they walk. Faster gaits are saved until they've got "being a burr" down.
Is it safe? Safer than skating or gymnastics. A LOT safer than driving in a car. But no sport is truly safe. You get hurt from time to time in any sport. Is it a dangerous sport? Not when taught correctly, and not at lower levels. Like swimming & diving. Little kids in swimming lessons don't get splattered cliff diving, because they don't DO that level. They're in the shallow end closely monitored. Same with riding. I've had my own son riding as often as possible (not a lot, though, time and money issues) since he was 3. I feel bad for him, since half of his friends have had their own pony since they were 3-5.