Yes, it's possible the turn signal switch is failing, but it's most likely still in the turn signal flasher. The right side bulbs might have enough resistance to heat the flasher and get it working so when you switch back to the left side the flasher continues to work. Before you buy anything, try switching the turn signal flasher with the Hazard flasher. That flasher will flash even with no bulbs. The fact the brake light goes out is showing the turn signal switch is working to some degree. It switches the power coming from the brake light switch and sends it through the turn signal flasher. If switching the flasher works, try using a TRIDON #EL12 flasher unit.
If it only happens when you have your foot on the brake, then you have a ground issue.
Also if you installed any (front or rear) of the turn signal bulbs with LED bulbs you need to install a Flasher designed to work with the LED bulbs (like a Hazard flasher). The LED bulbs draw extremely low current and will not heat up the factory designed #552 turn signal flasher.
Hope this helps