There are three wires going to the red taillights. (Plus a light green to the white backup lights that don't seem to be your issue.)
- Brown wire runs from the lights switch to all four red tail lights. That wire uses one of the two filaments - The brown is the +12V and the light's ground are that filament's connections.
- Dark green wire runs from the turn signal switch to the two right side tail lights. It applies +12V to second filaments of those right side bulbs. Again the socket ground completes the circuits.
- Yellow wire runs from the turn signal switch to the two left side tail lights. It applies +12V to second filaments of those left side bulbs. Again the socket ground completes the circuits.

The test I mentioned would verify the +12 from the actuated brake switch makes it to the new turn signal switch. (You had mentioned that all the other lights; turn signal, flasher, and lights were all working as was the switch so the only that white wire from the brake switch and turn signal switch had not been checked for continuity.