I have a few lines of thought here.
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School horses can be stubborn and difficult since they have to deal with lots of green riders pulling, kicking, and flopping around up there. Most are saints, but once they get to the “screw this” point, they become dirty stoppers, buckers, etc. Personally, I would only put an experienced riders on a dirty stopper, which brings me to my next thought.
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I have no way of knowing what level of experience your daughter has. If she’s coming unseated frequently, it suggests she does not have a secure seat. When she first started riding, did the trainer put her on a lunge line so she could develop a feel for proper seat and balance? Your daughter sounds like she could greatly benefit from a stronger foundation before moving onto cantering and certainly jumping. At the very least, I would ask that she be on a horse who show her the ropes until she is very comfortable up there. Until you can walk/trot/canter well, jumping should not be in the picture.
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With beginners, there seem to be two schools of thought. Some barns do lots of what are callled “up/down” lessons to help the new riders develop a feel for the horse. They learn how to post, leg placement, etc on a lunge line so they don’t have to worry about steering or even controlling the horse. They make sure the students can W/T/C at all three gaits, before they even ride off the lunge line.
The other extreme is the barn that puts your kid on a horse, hand them the reins and tell them to kick to go and pull to stop. These are kids that usually end up falling off a lot and most quickly lose interest in riding. While these are extremes, it does sounds your daughter is at the latter barn to an extent.
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As for the actual fall, if the head hit the ground at all, riding is done for that day. Get her checked for concussion. If she merely fell off and was just dirty but not broken, remount and move on. When in doubt , a visit to the urgent care center is never a bad idea. If your daughter continues to ride, you will start to see a pattern with doctors rolling their eyes every time she asks “when can I ride again?” While I hear your desire to run to her when she falls, she sounds like she’s handling it just fine. Riders are a tough bunch. You mention she is shy. It will definitely help her find her voice. I’m sure you will see a positive change there. That all being said, her first lesson should have included how to fall safely. Always tuck and roll. It greatly lessens the impact and would have prevented a broken wrist. I’m guessing she tried to brace herself on landing.
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If you’re watching the lessons and wondering if you have her with the right program and instructor, here are some things to look for. Does she look secure on the horse? Is she sitting up tall or is she slouched over and grabbing mane to save her balance? Is her lower leg still with her heels down, or is it dancing all over the place? Is she riding better at the end of her lessons than when she first got on? Watch lessons at some other barns if you can. You might notice some things for better or worse in comparison to your daughter’s lessons. Most kids want to get to the jumping part, but she’s young and has plenty of time to get there.
You’re a good mom for asking these questions. Sorry your daughter got hooked on such an expensive sport!