This seems to only happen in very narrow rivers: I’ve found that rivers sometimes are split into two different areas when crossing under a bridge - two closed and disconnected areas, but still connected with the waterway line.
Two examples I found (bridge way in the link):
Is this correct mapping?
Personally, I would say only split them at a bridge if the river name changes at that point, or it changes from a river to a lake or similar.
I would add the situation if the river/stream flows through a culvert below the streets. Those sections I usually do not map as area.
I think we have to distinguish between “splitting” and “ending in front of the bridge and starting again after it”.
Simple splitting is a natural thing to do if the area needs to be split because of size. The two areas are joined directly under the bridge to “hide” the boundary between them.
Usually a bridge is an object over a waterway, which doesn’t touch or alter it and hence there is no reason to explicitly not map the area for a few meters.
So, the second is wrong IMHO, unless the waterway changes properties under the bridge.
If the waterway changes, e.g. to tunnel=culvert, then obviously the area shouldn’t be mapped.
As another option, we could map the area under the bridge as a separate area with other properties - e.g. covered=yes, or (if they exist) additional tags for the construction of the riverbanks.
I don’t like the mapping in either of these examples as they show an unaligned gap, which looks ugly. Splitting the water areas like that only makes sense if someone also maps the area of the bridge as man_made=bridge:
If the bridge is simply a linear highway with a bridge tag, I would keep the water area independent and continuous.
(By the way, how did you get the maps to display inside your post?)
I would say that also then it is preferable to map water area under bridge.
On openstreetmap.org, click on Share on the right side, then HTML to give you a snippet of HTML with an iframe:
<iframe width="425" height="350" src="https://www.openstreetmap.org/export/embed.html?bbox=-113.50760936737062%2C53.54093220424611%2C-113.5016816854477%2C53.54394137728667&layer=mapnik" style="border: 1px solid black"></iframe><br/><small><a href="https://www.openstreetmap.org/?#map=18/53.542437/-113.504646">View Larger Map</a></small>
which when pasted directly into a comment without any formatting, gives you:
That’s the right answer; the water area doesn’t stop at the bridge, and layer=1 on the bridge ensures it will be rendered on top.