The best camera for beginners shooting on horseback

Christmas is coming and I want to buy a camera for my girlfriend, since she spends a lot of time with horses and uses my current camera to take pictures. The problem is that my camera does not take photos fast enough when I'm in the sports setup (it takes some photos and then it stops). What would be a good camera for beginners that would be good for this job?
Most of the images will be of horses trotting and galloping.
My budget is around £ 500- £ 600.
Thank you