How did Vince and Lyle's scam work?

If I understood Vince and Lyle's strategy correctly, they were swapping low odds horses with high odds horses between stables, and then placing the bet on the horse with odds against it.

The question is, how did they get away with it except for the one instance where they blackmailed the racing commissioner? Until they could blackmail him, he would have gone about his business checking tattoos inside the horse's mouth, and even without that, surely the jockeys would have been able to recognize some of the horses individually.