What should you do if a horse rushes across a bridge?

A horse that rushes across a bridge rather than walking deliberately has not yet developed genuine confidence on the bridge — it is getting across as quickly as possible to exit a situation it finds uncomfortable, which is a different thing entirely from accepting the bridge calmly. Punishing the horse for crossing, even hurriedly, is counterproductive: the horse made a forward try and crossed the obstacle, which is the direction the training needs to go, and adding a negative consequence to that forward movement teaches the horse that going forward is wrong when in fact it is correct. The problem is the rush, not the crossing, and those require different responses. The correction for rushing is to rebuild the approach and surrounding steps rather than the crossing itself: ask for pauses before the bridge at a comfortable distance so the horse learns that stopping near the bridge is safe and relaxed, slow the approach pace significantly by working on calm walk approaches from various directions, and reward heavily for any reduction in speed during the crossing compared to the previous attempt. As the horse becomes more comfortable with the bridge environment — approaching it, standing near it, pausing just before stepping on — the rush almost always reduces on its own because the urgency that produced it was anxiety rather than disobedience. Once the horse is approaching and entering the bridge with a slower, more deliberate step, begin asking for a pause with one or two feet on the bridge and rewarding the ability to stand there before continuing. Eventually the horse that was rushing will be the horse that pauses mid-bridge on request, but that outcome requires building from the approach rather than trying to slow the crossing directly.