Pasture means absolutely NOTHING happens to a horse when you take a turn. They don't age, but they don't become rested. So putting them there is literally like freezing time for them. Like, if you broke your arm and someone froze you for a hundred years right after it happened, you'd still have the broken arm when they unfroze you.

That's a ridiculous example, but it works. So basically, if you want to train the horse, they must be in the stable. There's no way around it. You could try saving money by only having a small handful of horses.
