Somebody brought up Bran being strapped to Dancer as an example that George R.R. Martin doesn't know that much about horses.
I actually think he does reasonably well. He knows what kind of horse people would use in different situations and climates, for one thing. But there are a couple of things about the entire Dancer thing that are wrong. The first is that you should not ride yearlings in the first place and should not put a disabled rider on a green horse (unless, of course, the disabled rider is an expert).
The second thing is strapping somebody into the saddle. Barrel racers sometimes do this with little kids for "safety." I've also seen "seatbelts" used in pony rides.
This is generally considered unsafe and wrong by many, but not all, horse people. In some cases the straps are designed to break in an emergency, but a lot of people feel it is still too dangerous.
Some therapeutic riding barns use these, others think they're terrible. (No doubt the ones that do are where Martin got the idea).
The worry is, of course, that if the horse were to trip and fall the child would be much more likely to end up under the horse. I will note I'm personally not a fan of this. But your characters might be - and seatbelts on pony rides used to be very common in the theory that then you don't need a sidewalker to catch the child if something goes wrong.