IMHO, rivers are not implemented that good. In order to be interesting and realistic, they should both help and *obstacle* movements.
Let's leave untouched the bonusmovement if unit travels thoru the river.
In all other cases, a river should then increase the movement cost of the terrain it is on, e.g. adding +2, +3, or +4mp, or multiply x2. In my opinion the best should be +3mp. [I'm speaking from the POV of a LT player with x3 movement]
So, to reach/cross a river is very expensive in terms of mps, but to leave it, it is not. Navigate through it will be as with roads (normal mechanics in LT).
This way rivers will become much more important in military strategy and they will become natural borders between the two shores, even more than hills, and as much as mountains. Rivers+hills or river+mountains will be important strongholds. To control this new borders (the river) will become very important, not only because they have defense bonuses, but because you don't want the enemy to gain control over them.
In terms of realism, it will add a lot to it.If you're trying to CROSS the river, it is obvious, you spend lots of mps. If you want to reach it, it may not be THAT expensive (you may just use your last 1 mp, after all), and you'll gain "control" over it. When you already control a river tile and just want to leave it, you're supposed to control both shores (as when you are on a mountain), so it will not cost you any additional mp. This makes a lot of sense.
ùTo build bridges should cost more mps than normal, let's say x2. Pillaging bridges when retreating, or with bombers should then become common warfare tactics.
Build fortresses on a river should also be more expensive than normal, say x2 again.