Best beginner camera for shooting on horseback

Christmas is coming and I want to give my girlfriend a camera because she spends a lot of time with horses and she uses my current camera to take pictures. The problem is that my camera does not take photos fast enough in the sport (it takes some pictures then stops) .What would be a good beginner who would be good for this job?
Most of the pictures will represent horses trotting and galloping.
My budget is about 500 to 600 pounds.
Thank you