2003 Ford Mustang Rear End Noise While Turning

Check at an auto parts store that rents or borrows tools to see if they have one called a "Chassis Ear". You can find them on Amazon too. It's a set of six microphones, a switch box, and head phones. You place the microphones near suspect parts, then switch between them and listen while driving. You can move the microphones around to zero in on the source of the noise. Be aware that most mechanics have never heard of this tool. Suspension and alignment specialists use it to find squeaks and rattles.

Differentials is a specialty area that very few mechanics get involved with. They require special tools, special gauges, and very precise adjustments to prevent them from causing noise or abnormal wear. Most likely what is making the noise is related to heat since it starts after driving a little. That would imply something is expanding and adversely affecting one of those critical adjustments. That would be found by painting the teeth on the ring gear with a special ink, then seeing how it transfers to the pinion gear and how it is removed from the ring gear. If it's ring gear noise, the tooth contact pattern would have to be real close to one side already instead of nicely centered. That contact pattern isn't going to change significantly on its own so if it's wrong, someone was in there previously and set it up incorrectly.

I doubt it's bearing noise because that would be there all the time. The axle and spider gears don't turn fast enough to cause an audible noise. Clutch plates normally chatter or moan when the whale oil additive is missing in the gear lube.

My best recommendation is to find a mechanic who specializes in manual transmissions and drive lines. Drive the car long enough to have it start making the noise, then have him drive it and listen to it. There's a good chance he will know exactly what is wrong and how to fix it.