Going to be a bit of supposition on my part, since I don't have either the radio or the mic and am having to rely on docs found online.
Turner wire colors I got from KC9Q in this thread:
https://www.worldwidedx.com/threads/turner-2-internal-wiring.133366/
White (audio) - Pin 1
Shield - Pin 2
Black - Pin 4
Set mic for electronic switching.
The reason for setting the mic for electronic switching is it is using the shield as a ground for the keying circuit. If the Turner diagram linked above is correct, when the PTT switch is pressed the black wire will be connected to ground if the mic is set to electronic switching. In relay switching, the black and the red wires would be connected together.
Also, the pins are based on the sister to this radio, the Pace 8193 and the schematic for that from Sam's CB 280. If it doesn't work with the black wire connected to Pin 4, check that the mic is set to electronic switching and then move it to Pin 3. Royce seemed to like having audio on Pin 1 and shield on Pin 2, so those shouldn't change.
In case I hadn't put enough disclaimers on here, proceed at your own risk. Since I don't have this particular combination of gear I can't say for sure what I've given you will actually work.
Good luck!