Hello,
I’m not familiar with the 5 wire picture you posted but i repaired a few turtle beach headsets in my days and if i’m not mistaken on your turtle beach …The red is right audio and green left audio. The copper wire is ground , the red/green is mic and the copper around that wire is also ground. You can solder a new 3.5 mm jack to these wires. The jack has four soldering points (starting from top to bottom) where 1=left audio 2=right audio 3 = the two grounds and 4=mic.
hope this helps,
Luis