Inexperienced Horse Rushing The Jump

I have recently started showjumping my 8 year old mare. She is relatively inexperienced but otherwise sound and very reliable. My problem is that whenever we go to a show (they are novice level) she jumps great in the first round but becomes hyperactive after that and will go through fences instead of jumping them.

The problem you have is that she is anticipating the jump. This is more common when you are against the clock and go slightly too fast for the horse. You could go two ways: Some people might put a stronger bit in and become more assertive; I would recommend you slow everything right down and go back to basics. Complete some rounds trotting until she can calmly complete several rounds. You can also give her more to think about, such as through pole work. In the long term, you might think about having her back teeth checked to make sure there is no discomfort.