Fixing a headset, TRRS
The easy way is to measure the resistance between the conductors. 

You need to prepare the conductors first. The thick white wire us most likely the microphone (+). You would normally use wire strippers to remove the insulation. Note that there is a possibility that the white wire is actually a shielded cable: white insulation over a shield over the center conductor.  But it most likely is a single conductor.  Strip it carefully to find out.

The other wires are most likely enamelled with solderable insulation. Use a soldering iron running at least 700F and put a small pool of solder on the tip. Hold one of the enamelled wires in the pool of solder. Add more solder as needed. 

Eventually, the solder will burn through the insulation and the conductor will be tinned with solder. Do all 3 of the enamel-insulated wires. 

Put the headphones on your ears and set your DMM to the continuity range. If you don't have a DMM, use a 1.5V AA cell. 

Touch the DMM leads (or battery) to 2 of the enamelled wires. You should hear a click from one or both of the earpieces.  Just a quick touch is all that is needed but do it as often as necessary to identify which earpieces are clicking.

Try all 3 combinations of the 3 wires. You will find out which is the left ear, the right ear, and which wire is common to both earpieces. 

The common wire is common for both of the earpieces and the microphone. 

Now simply connect the wires to the connector and try it. Do note that there are two distinct wiring standards for TRRS plugs. Tip is almost always the Left ear, the ring closest to the tip is almost always the right ear.  However, sleeve and the ring closest to the sleeve can be either Gnd / Mic or Mic / Gnd.

Yeah - the wonderful thing about standards is that there are so many to choose from [sad grin].