Fallon Taylor offers several great free barrel racing videos with tips on YouTube. Here is one of her helpful videos that talks about how to deal with a barrel horse that has a tendency to shoulder the barrels.
In the video she focuses on the second barrel. The second barrel is the hardest turn because we have to make a full circle turn around it. One of the first problems she discusses at second barrel is when riders don’t finish their first barrel properly and end up moving their horses over too far and going into second on the wrong lead This in turn causes the horse to change its lead in a difficult position making him come out wide on the backside because he has to take his room there.
Fallon Taylor demonstrates a few solutions to shouldering including counter arcing the horse away from the barrel, then pushing him past the barrel following up with finishing the barrel.
Here is the video:
As the barrel racing tips video continues she demonstrates where she likes to rate and position her horse going into the barrel for the most efficient turns (she likes to be 4-6 feet out and rate and stop when her knee is at the barrel). Fallon talks about body and hand position too. She suggests you sit deep and being sure to not pick up too early on your horse.
The bit she mentions and shows at the end would be a good bit (as she suggests) to use on a horse that has a tendency to shoulder.