That so-called 'instructor' needs to be reported. Definitely spread the word to anyone you know about her unsafe behavior with your little girl. You may help save other children from harm. Even though children do heal more quickly than adults, their necks aren't as developed as adults compared to the weight of their heads, so injuries to their necks can more easily snap them (if you want to freak yourself out, watch those car seat videos).
I've been riding since I could hold my head up on my own. However, my parents out me in front of them on their saddle. I can still remember thinking the latigo straps on the western saddles were my 'reins'. This was in addition to always being around the horses on the ground and learning all sorts of safety things at my parents' knees. Eventually, I was given the reins while mom or dad rode behind. Then I was given a fat, short pony to bounce around behind the big horses my parents rode. I think I was 5 when I rode that pony for the first year or so bareback because we couldn't find a saddle that fit him. I outgrew him by the time I was 8 or 9" so I got a 'horse-pony' of 13 HH that was an absolute babysitter. My little pony taught me balance and persistence since he was a naughty pony. My 'horse-pony' taught me everything else - she was awesome. In a perfect world, I think that's how all children should learn to ride.
Without horsey parents, the best thing seems to be lead-line rides for a long time with lots of ground interaction with a HUGE emphasis on safety. Then a graduation of sorts to longe line before finally graduating to independent riding on a reliable horse in a controlled setting initially at only walk and trot before finally easing into cantering and galloping. Then and only then is a child ready for independent riding in the 'real' world at all gaits. I don't think age has much to do with it except as a general guideline since every child is different and progresses differently. If you hold back a bright child with a natural incination to riding you're likely to sour them before they have a chance to really explore their talent. But if you push too hard too soon, you'll likely do the same or injure them. It requires a very talented and patient eye to bring on young riders - probably even moreso than it takes to bring up young horses.
OP I'm glad your daughter is safe. Take her to the nearest barn just for some horse loving, nose patting, and treat feeding. Tell her what a big girl she was to do what she did. Most of all reassure her that all riding is not like that and that when she's ready to continue, you'll find her a better place to learn. Then give her a great big hug from all of us here at The Horse Forum!
Advice is like a stall, sometimes you have to sift through it before you find something worth keeping. Manure happens, just shovel it!