ball joints or tie rods
A bad bearing can cause can the front end of car to make a squeaking noise when turning steering wheel either direction. It can also be caused by bad suspension.
Most likely it is low power steering fluid. Check fluid and see if noise goes away.
could be struts and or strut brg plate that mounts the strut at the top
The most common cause for a squeaking noise when turning the steering wheel on any vehicle, including a 2001 Toyota Avalon, is the belt that runs the power steering pump. If the belt is fairly new, a small drop of any dish soap on the belt while the car is running will clear up the noise. If the belt is worn, it needs to be replaced.
Any bushing, if its worn or old enough, causes suspension squeaking at low speeds and when going over bumps. The typical cause of squeaking noises when turning the steering wheel is low power steering fluid. The power steering fluid reservoir should be checked and the fluid topped off if it is low. Fill your reservoir with manufacturer-specified fluid (or have your oil-change service station do it). With a very low fluid level, the whining sound will be accompanied by a difficulty in turning the wheel.
The problem is most likely a loose belt, this will happen when steering piston and pump is under a load as in turning. More torque is required to spin pulley.
Worn ball joints?
If the sound occurs only when turning, (left or right) the most likely cause is a loose power steering belt. First check to see if the belt is tensioned correctly, or in need of replacement.
Many things can cause that, the simplest cause being lack of power steering fluid.
A bad alternator. Turning the steering wheel all the way is very demanding on the power steering which draws power from the engine and can cause the idle to dip down. A weak alternator is usually the cause for the stalling.
Low steering fluid can cause a steering wheel to whine when turning. A worn steering belt can also cause a steering wheel to whine when a car is being turned.