I have been hauling a trailer for years, first a stock trailer and then a 2 horse straight load.
As some have mentioned, I had heard that the horse must be hauled on the left side, because of the slope of the road. So I always hauled that way. Then one day I mentioned about always hauling on the left side to the trailer company, and the owner said that is not true. So I tried my horse, a 16 hand draft cross on the right instead of the left, and to my relief there was absolutely no difference hauling on the right or the left.
None. No difference. It hauled exactly ther same. The horse acted the same, except he was surprised for a second when I turned him towards the right. Unloaded fine, reloaded fine.
My trailer has a 7'6" stall to the chest bar, and another 3' for the horse's head, total 10 1/5 foot long stall, 7'4" tall, and a 5' long loading ramp. Never had a problem hauling any horse in it after they were loaded the first time. Plus I drive 65-75 mph on the highway. Maybe even 80 at times. Up and down big mountains. I couldn't do that in my stock trailer!
Also, most of the people I know haul 1 horse in a two horse trailer. Most of them would never load a horse in a slant load. But to be fair, the horses are large warmbloods, 16+ to 17+ hands. Never a problem, to load one or two.
Of course, the trailers are large, stable, well designed trailers like a Sundowner, usually a gooseneck, and the trucks are a minimum of F250 often a diesel.
Bottom line, use a truck and trailer designed well, and there will be no problems. Trying to sqeeze a large horse into a little trailer and pulling it all with an undersized truck - a disaster.