If the trainer you are working for is not a rider himself, his opinions and advice about riding is always going to be questionable. I think this is where you are.
IMO, how a racehorse goes under saddle, on the track and in an arena, is dependent on how you ride him. If you are a “big boy”, you can try to out muscle him, “hold” him with sheer strength. Some exercise riders are successful with this. But if you are not fit, not strong enough, you will lose this battle. The other option is “finesse”, which is the more civilized option.
Like any other horse, you want the horse to relax, understand the training, and enjoy his job. Riding softly will do this for you. Yes, you take the cross, but you don’t take a strangle hold. Taking a cross can still be soft. The horse takes the feel, not you. You just respond. Trot as far as you can, a good warm up is important. Pick up the canter, keep it soft as long as you can. As the horse begins to take more hold, you slide your cross down his neck towards his withers. Hold the cross with your hands only, not your arms. Save your arms for later. With the cross, he pulls against his own neck, not on you. He should be comfortable with the hold, not tossing the head (head tossing often indicates dentistry work is needed). Do not ride with “short” stirrups for regular galloping… the longer you ride, the longer you ride. You will probably be more comfortable with shorter than hunter/jumper length, but not “jockey” length. Your leg on the horse will give you stability. When you get to the point of “working” a horse (timed workouts), you will find it easier with shorter stirrups, but not until then. Shorter stirrup length is necessary if going to the starting gate for schooling, you have to have your feet higher than the bumpers in there. If you take a big hold on a horse, he will take a “big hold” back on you, and it becomes a strength of will contest. If you don’t take that big hold, many horses will not take a big hold on you. Instead, you can float along softly for an extended amount of time. At some point in the gallop, the horse will feel that it is time for a bit more of a gallop, and you will pick up some speed, which is OK. An open gallop is the result, and if you are on a racetrack, that is the place for this. You may not be able to “stop” the horse at that point, but you are not out of control, and it is not “racing” speed. The horse will ease his speed when he gets to the backstretch and he is ready to pull up. You need to know where you need to be on the track, depending on how fast you are going, and what you are doing. Fast horses are on the inside rail, or close to the inside rail. Regular gallopers are in the middle of the track. Trotting and warm up is closer to the outside rail. Walking home is coming the opposite way on the outside rail. If you are passing another horse, you do it on the “inside” side of that horse. You need to be aware of other horses around you, coming up behind you and nearby. You need to know who is on those horses, and what their skill level is to know how safe you are in their presence.
Yes, you may hold some mane, or the martingale yoke. The martingale yoke is known as the “chicken strap”, which I always found amusing. Yes, it can help, on occasion. Whatever makes you happy is acceptable. Until you get onto the track, riding young race prospects in an arena is not much different from riding hunter/jumper/event horses, IMO. Presumably, you know how to do that already?
Riding in company with skilled and experienced exercise riders is the way to learn many important things about galloping racehorses. With green horses, it is imperative that this happens, both for you and for the horse because the horse needs the experience too. Your trainer should know this. Your trainer should know how to adjust tack correctly, but there are trainers out there, with trainer’s licenses, that I, personally, would not trust, nor work for, and have no respect for. IDK if the fella you are working for is one like that, or not. You have to decide. Your life and well being depends on your ability to judge the information you are getting.
Getting information on galloping skills at the track, finding a good exercise rider to give you this information, agree to gallop alongside you, can be difficult. If they feel you are too green to be safe, they will not want to ride alongside you. They will just want to stay clear of you until you get hurt and go away, just to stay safe themselves. They have seen many green riders show up over the years, and have seen the outcome, and just wish to not become involved. There is little reason for them to help you, it ain’t like they are being paid to teach you. If you can worm your way in with a good one, be grateful.