Loug squeaking noise while turning

Have you checked the clearance between the brake heat shield and the rotor?
They are prone to bend and make contact with the rotor and make a aquealing noise kind of like the noise the brake pad sensor makes when the pads need replacing.
Pat special attention to the area between the lower ball joint and the rotor as it is a very tight fit.
Also, remember that when you bend it in one place it bends the other direction somewhere else.
So, it takes a little patience, but it is important as it is not good to groove your rotors.

Let me know if you need any more help with the issue.

If the heat shield is within a few millimeters of the pad, it can tough it in a turn as the loads and angles in the suspension change. Things flex a lot when making a turn.
However, if you have, and it sounds like you have, checked the brakes, the next things to look at are;

The power steering belt may be loose or failing as demands on the power steering pump are higher when turning.

Suspension items; Tie rod ends, Ball joints, (generally lower ball joints fail before upper), worn out bushings or loose components.

Sometimes wheel bearings can fail and cause a noise like you are describing.

The biggest item to check is the power steering rack itself. It could be loose or failing. If it is failing the fluid in the pump might smell burnt or might need to be filled every so often as the rack is leaking. Rack and pinion leaks are are to detect as there is a boot to protect the shaft that passes by the main seal. This will often hold fluid and make leaks hard to detect when looking under the car or where you park it.

If you have not rotated and balanced you tires in the last 3000-5000 miles, it would be a good idea to do that and get an alignment. While the repair facility is aligning it any loose or worn components will be evident as it will not line up properly if anything is loose.

Please let me know if I can be of any further help.