That depends on the state. In Maine, for example, a driver is supposed to make a right turn and approach while as close as practicable to the right curb, which means merge first into the bike lane. Here in Georgia as well you turn right from the right-most lane, including the bike lane for as much as 200 feet.

Evidently many motorists are confused by bike lanes at intersections and unless they know the specific law in their state it does seem contradictory. Right turns, generally, are safest when you don't leave room for another vehicle on the right, including a motorcycle or bicycle. Ideally the bike lane should be between the right-only lane and the straight through lane, although the transition can be tricky.