Sorry to sound a bit of a meany (I'm really not) but are you sure you're cut out for riding horses OP? You honestly can't think like this around horses. You will fall off again at some point, get your toe trodden on, be knocked over, get a broken nose from an over friendly horse etc etc. It's all par for the course. You just accept its a danger sport. Except in very rare and exceptional circumstances would you take legal action. Sad sign of the times, and its no wonder so many riding schools are closing down.
I will run contrary to the comments above - please no hate
The owner of the horse should have done her homework and supervised your first couple of times riding. We know our horses very well and what type of person they are suited to so this owner would have known, if given the chance, the horse would take advantage of a rider who is perhaps not as good as they think they are (which are quite a few of us lurkers on the forums). Most of us think we can ride better than we can. Coming back into riding after a 2 year gap is not huge, there will just be lost muscles and muscle memory. If I was to have a sharer on my horse, I would want to see them ride in various situations before I would agree to a share at all. Not commenting on the financial side. No claim for you I am afraid, you took a horse out that you knew had a control/bolting issue.
I'm not exactly sure what the OP is thinking of getting compensation for. I do hate this culture of 'it's not my fault, I'll sue' for every little thing that doesn't go according to plan.
It does sound as though the OP is not as experienced as they think they are but I also am concerned that the owner of the horse did not appear to take any steps to determine how good a rider the OP actually is and whether they were a good match for the horse.
Hopefully the OP was not too badly injured and has learned a lesson.
We also only have one side of the story, so there is that.
However, from what we know from the OP, I think both are at fault.
Most experienced riders also would not take a horse out into open terrain that they otherwise could not control/school properly in a confined riding space. So there was a clear lack of judgment there.
OP - you had misrepresented your ability. You would need to be upfront about that when enquiring about a claim, as I would suspect that you have only ridden at riding schools or on extremely quiet horses before. The language you use to describe your incident is not that of an experienced, competent horse person, but that of a novice riding school rider.
Then I would suggest perhaps finding a new hobby. As said above, with horses, accidents happen. Regardless of whether you want to carry on, I suspect that if you successfully sue for compensation, the only way you would find a horse to ride after that (incl at riding schools) would be to buy your own. No one else would be willing to take the risk of being sued, as I am sure you can appreciate.