Some thoughts as a USHJA certified trainer and an instructor of 25-plus years experience: My big concern here is that your daughter wasn’t able to stay on at the walk/trot when the horse stopped and dropped its head; understandable at the canter, but if it was occuring at a slower gait, then something is missing.
My biggest concern is the IEA–I’ve been involved with the IEA. Lately it seems that instructors have been putting students that are not ready into IEA classes/levels they should not be in. IEA, if you read the requirements, have pretty strict ideals for the levels riders need to be at. For example, a rider needs at a bare MINIMUM to have ridden at least a year under instruction to even be considered ready–and that usually doesn’t mean “off and on”. A good solid year of continuous instruction and riding. In addition, to be jumping at 18" in IEA you are supposed to be jumping 2’ 6" at home on your regular horse, to do 2’ you should be doing 3’ at home. I find that many barns are pushing kids into IEA before they really are ready–it is IMPORTANT that the student truely be ready–they are riding strange horses in a competition setting without being able to ride or school them before going into the classes–that alone worries me about your barn.
As far as the fall, I would guess if she fell when the pony put its head down she was pulled off–hanging onto the reins and following the head down. This is assuming it was just a head down and not a drop the shoulder and spin. In the case of the head just going down she needs to be shown how to slip her reins and stay sitting up, then gather the reins back up and regain the horse’s head. This can certainly be practiced by the instructor with the student–I do this all the time. Stand in front of the child, have the child hold one end of the reins while the instructor pulls on the other end the way the horse woud–student practices letting reins slip while sitting up, then practices shortening reins as soon as they can.
As far as practicing falling–we teach all students an emergency dismount. If they start to fall, they can use this motion to help them land on their feet in a safe manner. It will not help with all falls, but does help with confidence, and with them being able to safely dismount if they are so far off balance they are going to fall.
Finally, as far as getting back on the horse where the traumatic event happened…as an instructor, I would not put her right back on that horse after a broken bone like that, or even if she had that many falls and was traumatized by it with no injury. She needs to regain her confidence, and that is not going to happen on that horse. Not saying she should never ride that horse again, because if she continues to ride, and the horse continues in the program, and doesn’t really have a major behavioral issue, at some point she should ride it again so she knows she can–but that may be quite a while down the road. Putting her immediately back on that particular horse is not going to help her–she is going to be too nervous and not ride well, and may fall again just due to the sheer panic of being on that horse, which in turn is going to create more problems. She needs time to improve her riding, time to improve her confidence, etc. Any decent instructor should know this. And she is probably going to start back at a lower ride level due to nerves and need to be built back up in confidence, unless she is an unusually brave kid. But I would not expect that response out of a kid riding off and on for two years.