I’ve had this issue too, since my favorite commute route involves taking 17th to L to 15th. WABA has a Cycletrack 101 page that gives the following options for turning right from the L St lane:
-Wait for a pedestrian signal and cross traffic in the crosswalk.
-Maneuver in line with the traffic waiting at the cross street. Proceed across the intersection when the light changes.
-If you are comfortable doing so, before reaching the intersection, merge into the main roadway and over to the rightmost lane, then turn as normal. Remember to yield to oncoming traffic and be safe, if you choose this method.
I’ve personally found that the last one works best for me at 15th St — so I just basically stay in the right lane for the couple blocks that I’m on L St. If you’re on L St for a while, though, maybe it would be best to wait til you hit your last red light before 15th and use that as an opportunity to get over to the far right lane?