Most problems with barrels are bc the rider lets the horse take over the whole job. Stop running barrels this WHOLE season. Instead, school on the flat for total obedience. Teach him to take cues between gaits and cues within gaits. Get a book on basic dressage or take a few classes in it so that your barrel horse gets some education as a RIDING horse.
The reason the horse flipped is bc he/she is running away from that whip SO FAST that the horse wasn't paying attention to where the feet go. People can trip down a flight of stairs doing the same thing.
When you get back to the barrels again, make a bunch of exercises out of it. Walk the barrels, go from #1 to #2, then back to #1, then to #3, then back up, then circle ALL of the barrels, change gaits between barrels--just get creative. Horses don't forget ANYTHING, and he won't forget how to barrel race just bc you took some time off. Once you have schooled your horse like this you're going to feel more like you are driving a tuned up sport's car, and they aren't always driven at race car speed.
I have a huge issue with this post.
First off, you are assuming this horse was not a riding horse first. If the OP is working with a trainer, then no self respecting barrel horse trainer would let her get on a horse with no training and just whip and spur around the barrels.
Doing the dressage work is an excellent idea, however the OP doesn't sound like she has a problem with going slow. She has a problem with going fast. I'm going to quote 1RedHorse here: If the problem appears at speed, fix it at speed.
Now, this doesn't mean fly around at mach 20 all the time, but I'll talk about what I would do with my lesson kids in a minute. The same thing happens because I don't see how doing transition exercises is going to help her with her problem either...Unless, OP, your fear does also appear while going slow? In which case, by all means.
On the comment about the theory on WHY the horse the OP say crashed....There is NO correlation between using the whip/stick and wrecks. There are a million factors that could have caused that. I've had horses trip and fall on me at a simple lope because they just hit bad ground. I do NOT whip or even push going to the first barrel, (Free running horse), but my mare once hit a patch of deep ground and stumbled. I do not typically use a whip on my mare as she is a very free running horse, however when I do there is definitely no "fear", and you will find very few horses who are actually afraid of it. It's a cue, just like anything else, and you will find they are there more for noise than anything. Just like when people say barrel racers "kick" too much. So many of them FLAP their legs, but don't actually kick. You kick too hard, you slow a horse down. It just looks like they are kickin the snot out of the horse when they aren't.
Now, onto helping the OP. Since you work with a trainer, I'm assuming you probably have an arena with good footing to practice in, right? What I would do is take him around the rail of that arena, put your outside hand on the horn and your inside hand on the reins, and "breeze" him. Walk, trot, canter, then let him run. Push him with your legs and keep encouraging him to extend out without the whip first. See how that feels. It's a controlled environment, no tight turning, very safe on a broke horse. I do this on my mare quite often. Actually I think I even have a video...Our arena is pretty round but you can do it in a square or rectangle too.
Not one of our better days (She had a lot of time off before this) but for the sake of the thread I'll put it up, haha. It's the first clip. Notice how I don't push her too much. It's a confidence builder for you but also for your horse, showing him that he can run and it's okay. The only way they'll find their feet and balance for running is from preperation...And finally, just letting them do it and teaching them it's no big deal.
Also, when practicing your barrel pattern, emphasize the drive out of the barrel by speeding up significantly out of it. Here is a video of a lesson, the two palomino horses are good to watch and I think there's some good advice there from my own trainer. It's all about focus, forgetting your fears and just doing what you need to do. It's hard but as you get more comfortable you will find yourself tuned into your horse and know when you should ask for more, and when you shouldn't.
Finally, just go out and play sometimes. Have fun. If you find a big open area let him run. Just remember after doing all the running, emphasize your rate. It's so cool to see and ride a really awesome barrel horse, because you can go from a flat stretched out gallop to a pony lope in .2 seconds just by giving them their command to rate. You don't find that in any other sport, except maybe reining, which is why I condone barrel horses as reining horses first (But that's another story)
Just remember that falls are probably going to happen, but this is a wonderful sport and going fast is such a freeing feeling that if it's something you love, you can push past it. Good luck OP and let us know how it goes! We love updates here and would love to add you to our barrel racing "family" here on the forum :)