Engineer bridgades reduce but do not eliminate the river crossing penalty.
In standard HOI, however, leg inf including moutain marine and para have zero river crossing penalties without engineer brigades when two river crossing techs are researched. Which isn't too realistic but that's how the game handles unless you mod it.
For motor mech and pz div you need to do all river crossing and wading techs to end up with a zero river crossing penalty. The river crossing penalty is 50. The eng bde is +30 while two inf and three wading techs add 20 resulting in a penalty of zero for these units when they have an eng bde. Without an eng bde or without engineer leaders, these units get a -30 river crossing penalty.
CORE handles this differently and still give some speed advantages to eng bde.