бежа́ть -- unidirectional, one-way - "to run, escape, flee ; to avoid, to shun"

бегать -- multidirectional, return trip - "to run, jog; to go/visit often"

Gallop is one way a horse runs. Horses also "trot" and "canter".

I recall a local horse-training who taught riding and also rented out horse by hour for locals to ride. He was adamant that his horses could not be ridden at more than a canter. If a rider galloped a horse, the riding session was over, and the person was banned from ever renting a horse again.

Ah, I understand the confusion now. Бегут is simply an irregular conjugation of Бежать, not of Бегать (which would be бегают).