Prepositions – “Under the Water” vs. “In the Water”: Proper Usage

This sentence is both correct and makes perfect sense.

"The water had overflowed" is, on the face of it, fine. Overflowed means to spill beyond capacity, and that's definitely a thing water sometimes does.

It's often followed by a description of what was overflowed, but there's no rule that says that has to be the case. Instead, here, the prepositional phrase "under the bridge" describes where the overflow happened.

Since it is rather normal for water to be under bridges, that's not at all surprising. More context would make it completely clear, but without that, I assume that a river or stream overflowed its banks below a high bridge.

Like this:

Or, alternately, it could be a bridge over a road, and water may have overflowed onto that road (or, using a slightly alternate meaning of "overflowed", simply overflowed the road). Like this:

You can say

stand under the rain

and it will be understood, and people will realize you are a non-native speaker.

stand in the rain

is used since the rain kind of surrounds you (it falls on and around you), whereas

under threatening skies

firmly places you below the sky.