I am compensatiing with moving subjects to have them nearly out of the frame to get them closer to the center of the frame.
I presume the horses are running from left to right, and that you press the shutter when they are in the center of the frame.
1) Are you panning the camera left to right as the horses run by?
2) Are you "following through" with the pan after you press the shutter?
If you are doing both of these things, the only way I can figure that the horses would end up on the left side of the frame is that you are panning at a faster rate than the horses are moving. In the brief instant after you press the button and before the shutter releases, the camera continues to move to the right faster than the horses have. This assumes the horses maintain a constant speed.
If you are panning, then stop panning when you press the shutter, then the horses should be right of center.
If you are panning at the same rate as the horses move and are following through at the same rate, then they should more or less be in the center when you press the shutter.