I am of the opinion, based on the age of your child plus the fact that she has been thrown twice now, that the horse needs to find a new home pronto.
Your daughter is way too young to have to deal with something like this, if she is not already a very tough rider.
You have had this horse just the right amount of time for it to decide that your child cannot control it, and that it doesn't have to do anything it doesn't want to.
And much of this can be attributed to the horse attitude, of "no I don't want to" rather than barn sour, or adjustment problems.
And too, if you or your daughter are babying this horse, or dealing with it in any other manner but very dominant and firmly? This is the type of thing that results from that handling.
Very rarely have I seen a 4 year old that I would have entrusted a child with. They are out there, but very rare. If a child has grown up in horsey family, and by that I mean parents that are trainers, or very well versed in horses, then a 4 year old might work, depending on horse and child too. But as a rule?
Nope. No way.
I just would not risk this anymore. Not worth the headaches. And sure not worth the hospital costs, or the funeral expenses either. Too many decent horses out there to mess around with this one.
Get some lessons especially on handling horses, from a good trainer, that is not going to overhorse your child when you look to buy, (and those types are out there), and then find a horse that is a little older, and less likely to try things.
But again, a horse can be ruined by someone babying them, or by a human behaving in a submissive manner.
Horses make me a better person.