Depends on how big the bridge is and what type you need.

A suspension bridge can be built by pulling ropes across the water. You don't even need to pull the main ropes across first, you can pull a lighter rope across then use that to pull the larger rope or join multiple smaller ropes together to make the large bridge.

You can also launch a bridge, that is, build it on land then push it out over the water. This is still an option for building bridges, sometimes by building a segment, launching it out then building another segment and repeating the process. You can do this with piers in the middle or as a single span.

You can lift the bridge out into place - this may need larger lifting gear but it can also be done in parts and then assembled in place. You can also use a crane on a boat to float the bridge out then lift it up.

All of the above can be done to completely bridge over the deep water if you can make the bridge long enough between supports. If you can't, then you'll need to put supports in. A coffer dam is a fairly common way of doing this which I see has been explained already. You can also use a drilling rig to drill a pier into the ground without needing to dive down but both of these methods wouldn't be achievable in very deep water without reasonably modern technology (post industrial revolution at least). Before that, they were limited in where they could build bridge piers to areas where they could build dams deep enough or where they could drive piles into the riverbed from the surface