Here's rivers in the game. Some are exploitation, some are not, so I really don't know what is the movement cost rule...

1. https://imgur.com/UmXZI9X untouched terrain, 1 movement point, ignore movement penalty

2. https://imgur.com/VViYE87 exploitation, base movement cost: 4

3. https://imgur.com/kapmnRe untouched terrain, movement cost: all remaining

4. https://imgur.com/R7XiDFr exploitation, base movement cost: 4

5. https://imgur.com/R7XiDFr exploitation, 1 movement point, ignore movement penalty

What is the movement cost rule for rivers here?

Second problem, I don't see any roads in this Open Beta. Is this only me or you guys experience the same? Visually I never see bridges in past OpenDev too. If the river movement cost is cut because of roads, that explains some of this discrepancy, but visually I can't see anything.

Going into a river if you're not navigating on it absorb all remaining movement cost. If a road go over it it should only cost 1 (like any other tile) So I guess your other cases are bug.