I had the same problem. And my best horse will not show up ing the racing page when I go to enter!

From the horse racing game page:

Note that your horse will only appear on the list if it has enough training to race and has trained this week.

The race results will be based on your horse's heart, strength, speed, mood,track conditions, and your choices during the race. Note that this game looks at traits related to Cross Country.This means you should choose horses that are training in Cross Country or Eventing.

From the game help section - linked to the horse racing game page:

Some Tips:
We recommend that you only race horses that doing very well in their level of showing, for the best chance to win. A good strategy would be to look at your Show Report and see which of your horses are scoring well in Cross Country. Also, your horse's mood comes into play. For example, if you have a hot or spooky horse, and you use the crop too much, they will be likely to spook and loose placement. Lastly, the track conditions may affect the outcome. A fast track favors a fast horse. A wet track favors a strong horse. A sloppy track favors a horse with much heart. And an average track favors a horse that is balanced in the cross country traits.

Why isn't my horse moving forward much in the race?
If your horse doesn't have much training for it's level, then it won't progress much. It is better to forfeit the race and try a different horse who has more training at their current level. Make sure you choose horses that have decent amounts of training for their level in Cross Country traits. If you choose a Dressage or Jumping trained horse, then it is not likely to do well at all. Freshly captured horses and horses that just leveled won't have much training at their level so won't do well either.