I have tried looking in the manual but I found it a bit unclear. Doe the rudder damage come into effect at the start of the turn after you get it and last for one turn or does it come into effect at the start of the turn after you get it and last for the rest of the game?

Ie. scenario 1
round 1: in phase two a plane draws a rudder damage right. The plane has right turn i phase three but it gets played as usually.
round 2: the plane may not make any right turn
round 3: the plane may yet again make right turns unless another rudder damage right was recieved in round 2.

or scenario 2
round 1: in phase two a plane draws a rudder damage right. The plane has right turn i phase three but it gets played as usually.
round 2: the plane may not make any right turn
round 3: the plane is still forbidden to make any right turns and will keep being forbidden.

Your first Scenario is right

Ah great. We interpreted it that way after my wife got both left and right rudder damage in one attac. On the other hand I got a four and an explosion in the same exchange
That taught me that when you fly the lighter play you don't go head to head with the opponent