I never understood why they changed the connections of the joystick port midway through the life of the system. WTF?
Anyway you can get a 6-button controller to work on the TurboGrafx by taking an existing Turbo Controller apart, removing the wire (it plugs right in to the board, no soldering/unsoldering necessary) and putting it into the 6-button controller. Be aware, however, that not all TurboGrafx controllers are equal. They do not all connect to the board the same way as the 6-button does. It may fit, but it may not work. It's trial and error.
But the good news is once you do get it to work, you'll never want to go back to the 2 button pad because the 6-button pad is a bit bigger and we Americans have gigantic hands.