They drained the water.

For building across a river you could wait until a dry season to build your bridge when the water level would be low or you could sink logs into the ground to make a sort of dam around your construction site which you could then drain(either using pumps or just buckets depending on technology) then you build your support pillar before breaking down the dam you built.

Mostly, it involved building in the safest and easiest place instead of the most convenient for travel.

TLDR: boats

Suspension bridges at a basic level only required a rope to be pulled across a river, and that can easily be done with a boat.

Larger footings could be poured from a boat as well.

While there's a lot of prep work and drilling down on a river bed for bridges today, it more ancient times they would likely have just floated a barge to the spot and dumped large quantities of gravel and large rocks to create a solid footing

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

There is some really cool engineering tricks involving a thing called a caisson.

https://en.wikipedia.org/wiki/Caisson_(engineering)

Basically it is a box to keep water out while they build the footings.

Build the box, pump water out, work in the relative dry while hoping pumps keep working and the box doesn't leak.

Another tool they used was driving pylons or piles deep into the ground. These could be put in place from barges or other floating platforms.

Another kind of bridge is like a rope bridge.

Send a small rope across with a projectile (or a person travelling the distance by boat and/or on foot) Use this rope to pull a larger rope across. Keep on bringing rope/cable across the gap until you have enough strength / support to build a bridge.

There are different types of bridges for different kinds of challenges.

There are many different ways to do this depending on the available technology, cost and the ground conditions. For example we know the ancient Romans had pile drivers on barges which they could use to drive long piles into the river floor that would support the bridge. Some rivers would also dry up for some parts of the year, at least enough to build a bridge. Or the rivers could be partially drained for the construction. But even with these different techniques available the most common was still ferries at river crossings. Bridges are relatively new most places, most of them less then a century old.

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

To add to the previous explanations - here's an animation how a 13-meter tall bridge in Prague was built in roughly 60 years (Charle's Bridge). It's an official reconstruction of the local historic society how it was built. Should answer some questions. IIRC it started 1348 and was opened 1407. (not sure about it and too lazy to check it)

https://www.youtube.com/watch?v=nJgD6gyi0Wk

Edit> 1357 - 1402, meeting was less important so I actually checked it :D

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I remember an architect saying:

The bridge is 20% a bridge and 80% how to put it there.

There are several ways, but can be grouped in:

Remove the water, literally temporarily reroute the river.

Redirect the water: build a dam around the future pillar location, drain the spot.

The floor is lava: rework the ground on the two sides to make it firm enough to place massive equipment, from which you launch the premade bridge over the gap.

The floor is lava but old school: you build a lightweight wooden/whatever bridge from the shore into the river, join with the same structure coming from the other side. Use this structure as a platform on which you build the bridge.

The pointy stick: you stick giant sticks in the water and put the bridge over those sticks.

The suspended thing: you build a cablecar-like thing, from which you deploy wire after wire until you have a “golden gate”thing. Then from those wires you work your way down to build the road part.

Ice is your friend: stick pipes in the water, run coolant in the pipes and create an ice wall. It can be done to help in any of the above methods, where applicable. Even to freeze the terrain to prevent water to filter into a hole/excavation/whatever.

In all cases, the problem is not being under water, most construction hate water or hate being jointed in water. So even if you are the chief of the scuba planet with millions of scuba soldiers, you probably want the structure to be built in a dry environment. Then you can flood part of it, but only after it’s assembled or cured.

The above methods can be mixed in many ways.

They damned and drained the water.

The equipment to go under water is a lot older than you think too. Check out diving bells for some terrifying nightmare fuel.

There are several answers here that are correct, but they used a different method from any of them when the built the Brooklyn Bridge. They built a wooden "room" larger than the bridge supports, and submerged it, and essentially just made two big square holes in the river to work in.