It is hard to say without seeing you ride. But it might very well be that you're trying too hard and that way, your body tenses up instead. And then the ride becomes bumpy ;)
Try to loosen up your hips and keep your lower half relaxed, try to just follow the horses motion, like waves in the sea.
Ideally, you will be able to keep your upper body straight, but your lower body (hips) will be relaxed and will move along with the horse. For better stability, you also need to use your core muscles while doing this, but that will come later I think. First just get a good feel of the motion and try to relax, breathe and let your body follow your horse.