This is one of the things that pisses me off almost every morning.
You can completely ignore the bike as it's not relevant. The problem with making a right turn in the city takes a stupid long time because you have to wait for pedestrians, which means you'll be making your right turn just as the light is turning red (or after).
It's aholes that don't merge into the curb lane to allow traffic to pass. They happily sit in the right lane and because they are forced to wait to turn, they now force EVERYONE to wait. Since they won't be able to turn until the light is red, everyone basically has to sit there until the next light.
In reality, as soon as some ahole starts to do this, everyone attempts to merge or "half-merge" into the lane to their left, cutting other people off and just making the whole thing a big dangerous clusterfk.