There are many ways to build bridges over water, depending on specific site conditions, technology prevalent in the country and technical capability of the contractor.
All bridges (except floating ones, which are rarely used permanently), need a foundation, which is rested on the bed. The supporting columns (known to bridge engineers as “piers”) are constructed over these foundations, on which the superstructure is finally laid.
Bridges over shallow waters
In shallow waters, foundations may be laid by temporarily filling up or enclosing the specific location, over which piers can then be cast. Alternatively, when the soil is not very ‘good’ in the top layer, temporary rigs are erected and piles are driven inside the bed.
The superstructure can then be constructed either by taking support from the already erected piers, or by temporary platforms / mud-islands in the water, or through barges (though rarely in shallow waters).
Bridges over large rivers / sea
Where the water body is large and the water deep-
a) foundation may be sunk inside the bed from the top
b) Rigs may be employed to cast / drive piles on which a cap is then cast to support the pier
c) A cofferdam (a wall enclosing an area inside a water body) is first prepared, inside which water is constantly pumped out and dry working conditions are maintained. The foundation is then constructed inside the cofferdam.
Once the foundation is laid, the supporting piers can be either cast in-place or cast in the yard and brought to site on barges and attached to the foundation.
For superstructure construction over deep waters, more often than not support is taken from the existing piers.
The bridge superstructure may either be cast / placed on a launching gantry or truss
Or it may be constructed in a cantilevered fashion starting from the piers
Or segments may be supported by huge suspension cables and stitched together after bringing in position-
These are the most common practices as per my knowledge, of course there are other methods like incremental launching, sliding etc.; bridge engineering, like most fields in engineering, is fascinating in the sense that there is no perfect ready solution to any problem, and engineers keep on coming up with innovative ways to do the same things better. But I hope this answer gives you some idea.
Disclaimer: None of these pictures are mine to claim, I have picked up all of them from Google searches as I was writing the answer.