Scouts will pole-vault across chasms where they cannot leap them. Bridge crews portage and push (lift! run! drop! push!) their bridges across.
The maximum chasm width a Sadeas bridge can cover is about 15', perhaps a bit more with the right staging. This doesn't sound like much, until you consider that the goal is to move an army. Even small gaps of as little as 3-5 feet need to be bridged when you're going to march a few thousand men and materiel across them and you don't wish to get held up.
During bridge assaults, multiple bridge crews are sent in at the same time in order to divide enemy response. Parshendi will fire at the crews with arrows, but they need to withstand covering fire at the same time (with little coverage of their own) and they are not carrying a wealth of arrows (fletching and shafts being hard to source on Roshar, especially without soulcasting) so they do run out of ammunition.
And still, running a bridge crew is considered a death sentence. It's incredibly dangerous, deadly work. And it is meant to be so, Sadeas uses it as a punitive threat for discipline in his own warcamp and considers the bridgemen to be human shields, absorbing arrows that might otherwise be shot at someone useful (by his standards).
The main advantage to a Sadeas bridge is speed of deployment. He can get his troops out and across the Plains faster than almost anyone else using any other method (Dalinar's siege-bridges are much safer for the troops and engineers, but they advance at a painfully slow pace). And this speed of action is of the greatest importance, when you consider that the goal of the Shattered Plains War is not the elimination of Parshendi, but the acquisition of gemhearts.