What you could do is make it appear that there in no river there, but hide it under all the overlay graphics.
What you do there is make the hex where you want a bridge to go become a clear hex, once you have generated the tiles (start it as sea). Set a major river to go over that hex, and the two ocean hexes adjacent to it. Then put the road/rail over the top. Should look something like this when you press the spacebar in the editor;

- bridge.JPG (91.29 KiB) Viewed 1646 times
But in game, it appears like there is just the ocean and bridge:

- bridge ingame.JPG (120.26 KiB) Viewed 1646 times
.
also, you have to use a major river, not a minor one, or else you will be able to exploit the trick and move units from other sides (rather than where the bridge goes), onto the bridge, and then off it - without using the bridge.
That way, the bridge functions as if it was over the river, but the river itself is invisible (instead you are using oceans to represent it) - however the game ignores the overlays, so you see ocean, game sees river.
Hope this helps,
- BNC