If you’ve had prior experience with horses you’re a step ahead already! I don’t know if you have an idea of what kind of equine photography to aim for. Some specialize in portrait work, for using in ads. Others go for the action — rodeos, roping, cutting, showing, etc. Then work them for contacts. For example, if you want to shoot cutting horse competitions, the NCHA site has lists of all sanctioned shows. Go to as many of them as you can. Introduce yourself to the show producer or secretary, give him/her plenty of business cards, see if the club already has a photographer. The more you network, the more exposure you’ll get. Even take some shots if it’s a multi-day event, next day tack prints up around the concessions area (again with the biz cards) Also, be well-informed about what you want to shoot. If it’s a performance class, know what a well-executed maneuver looks like. A dramatic photo of a cutting horse blazing across the pen might look wonderful . . , except to those who know the sport and can recognize that the horse is way out of position!

that’s a lovely shot

I’m just an enthusiastic amateur (with cameras AND horses) but a couple things come to mind. The action on a racetrack makes continuous autofocus mandatory. Pentax — my gear — calls it AF.C, Nikon may term it something different. If you aren’t already using it, give it a try! That may help with the finish line.
Not so much with the turn; the focus mechanism has difficulty tracking movement directly forward) at the camera. With each shot it has to hesitate just a sec to recalculate.
Are you shooting at the highest ISO possible without generating noise? And at shutter priority, letting aperture fall where it may? I would guess that for racing shots the depth of field isn’t nearly as important as speed so opening it up would let you bump the speed up.
Hopefully others with more expertise will weigh in with better advice,

-pic for attention-
Image is of a 2x Olympic Gold medalist Buck Davidson on Copper Beach. Yes, I intentionally took this to focus on Copper and not Buck (sorry Buck!) https://uploads.disquscdn.com/images/58b8c2bfb6230b58e9e169369f6cc42c396553714e3158062e02803ce7f3c479.jpg

NEVER take pictures of a horse without the owners’ consent. Don’t pet them without the owners’ consent either. Horses can change moods in a split second. One minute the horse can be an angel and the next minute they’ll try to bite you. Talk to the owners – ALWAYS

Additionally…and I have seen MANY photographers do this at events and I want to lose my mind. If there is a jump, right next to a fence, do not try to get on the ground and shoot up at the horse and rider coming over it. That horse sees a monster hiding on the other side of that jump and many times will not go over. Now you just ruined that class for the both of them and cost them their entry fee.

ALWAYS stay on the side of caution. Remember you can affect their round by being too pushy to get the shot…even worse you can get them hurt.

Now I have gotten these pictures, however, as a rider myself, this was done with the collaboration of the horse an rider. Steps were taken to acclimate the horse, and it was done outside of an event with every safety precaution taken. However, I personally have been in the horse business 75% of my life.

always get permission from the horse’s owner before feeding treats however as some horses have allergies or are on special diets. Also if cut the wrong way (like carrots cut in discs) they can choke a horse. Slice carrots, apples etc lenghtways.