They did not build bridges over "deep and/or dangerous water" before diving equipment existed. That is bridges with support in the water. Rope bridges and other simple suspension bridges have been built over rapids if the distances were short enough.

If it is even shorter you can build a solid bridge that is just supported by the ground on the sides of the river.

In relatively shallow and nice water you do not need to go underwater you can still expose the bottom.

The simplest way to explain this is by building an alternative path for the water and then making a dam with dirt and rock in the river. The old river bed is now dry and you can work on it. Destroy the dam and fill in the digestion and you have a bridge. It might not be the simple thing to do for a large river but it was possible.

You do not need to do that for all of the river, build a cofferdam that encloses parts of the river remove the water from the side and you can work there. The wall of the dame can be large baskets you fill with rocks and dirt to keep the water, coffer is an old word for the base.

Or build a small coffer dam by driving wooden pillars into the river bottom to remove the water. It only needs to be just larger than the pillar you intend to build to support the bridge. This is still common we just use metal walls bridges and other stuff that need access to a river bed.

https://en.wikipedia.org/wiki/Cofferdam

Another way is driving a wooden pillar into the bottom with a pile driver and letting that support a wooden bridge that extends a bit out in the air so you can drive down more wooden pillars. Ceasar's army built bridges like that over the Rhine River in 10 days. It looked something like

https://c8.alamy.com/comp/BA805M/architecture-bridges-roman-bridge-of-gaius-iulius-caesar-over-the-BA805M.jpg

https://en.wikipedia.org/wiki/Caesar%27s_Rhine_bridges

So they built bridges over rivers with support in the river since ancient times but it was not very deep or dangerous water where it was done. It was quite shallow