Sharp panning shots are EASY...just shoot at a faster speed and pan perfectly in-synch with the subject. You can pan at ANY shutter speed you wish: 1 second, 1/3, 1/15, 1/20, 1/40,1/125, 1/250, 1/500. The "correct" speed to pan at depends on many factors. Subject, image size, direction of movement in relation to the lens axis, desired effect, and so on.
The idea with panning though is, quite often NOT to have it be "sharp", but to convey a feeling; a sense of movement; a sense of speed, or of motion...so...I'm not sure what the desired effect you were after was, but the idea that the horse is moving up and down is not a deal-breaker....it is in fact pretty normal. Most of the time when people use panning it's to give a sense of movement, but there are times when we pan simply because we MUST do it, just to get a marginally usable photo, like when you're at wide-open f/stop, max ISO, and the subject is moving fast....you pan then, and stop the still parts, and the moving parts are blurred.
Here's an article I think a lot of people might find useful, from a master photographer from National Geographic.
Jim Richardson on Panning to Capture Motion -- National Geographic