Do the effects of turbulence change when banking/turning compared to level flight?
Turbulence affects aircraft in the same way whether it is banked or not.  The difference however would be the end result of the turbulence.  

For example, if an aircraft were to encounter turbulence strong enough to roll the aircraft 20 degrees to the right, the end results would be a 20 degree bank if starting from level, but 45 degrees of bank if they were already in a 25 degree banked turn.  If this were to happen it puts them closer to their stall speed because of the higher bank angle, and if they were already going slow and the airplane is at a low altitude, it could be a problem.  

Fortunately, most turbulence strong enough to cause this while at low altitude is caused by the wake turbulence of other aircraft or microbursts, and we have pretty good procedures in place to keep it from being a problem these days.  If they aren't at a low altitude it matters even less because there is altitude for the aircraft to recover even if it does get into an unusual attitude.