Riding Safely Around Horses
1- when coming up behind a horse, even if it is not on the road, but beside the road (even if they're not right in your way, bicycles can still surprise the horse, which puts the rider in danger), make your presence known well before you pass. Don't ring bells or make a big clamor (this will make any spooking or jumping MUCH worse), but do consider shouting out "behind you!" or "bicycle, passing!" or something when you're still far enough away to not be hurt should the horse suddenly jump. In addition to warning the animal of something behind them, this alerts the rider so (s)he can be prepared to deal with any horsey shenanigans or warn you if there are likely to be issues.
2- slow down. Slower speeds allow the animal to get a better look at you. If you've announced your presence with a little shout or talked to the rider, the animal knows you're some kind of person, which should hopefully make them less alarmed about this bizarre thing that seems to run without legs. I know that if you're out training, slowing down isn't what you want to do, but it is the safest thing for everybody (this goes for if you are driving a car, as well- pass wide, and slowly. do not honk the horn, LOL)
3- If the horse starts jumping around, ask the rider how you should proceed. They may simply need a second to get out of the way and stopped so you can pass safely.