Thumping sound from 2004 camry, stops when turning left. increases at around 40mph.

Check oil level in the transmission. Also check if the floor mat under the accelerator pedal is not so thick to resist the throttle pedal movement.

Have the wheel bearings been checked?

As you know your Toyota is front wheel drive. Have your CV axel checked out.

if your lucky, its just a u-joint going out.

try replacing the u-joint which connects the driveshaft to the rear end(differrential), and if that does not solve it, then have your differrential checked out (a bad differential usually wont get all the power to the wheels that its supposed to) or just have it replaced, possibly with a junked differrential form a wrecked vehicle with low miles.

first rotate the tires see if there is a change. If that does not have a desirable effect there may be a worn wheel bearing,take note if the sound or vibration gets louder as you turn a corner, try left turns as well as right turns if the noise increases it is a wheel bearing.

IF IT ONLY THUMPS WHEN YOUR CAR IS MOVING YOU COULD BE WRITE. PAY ATTENTION TO THE THUMPING CLOSLEY, DO YOU FEEL A VIBRATION IN THE STEERING WHEEL, IF SO THE PROBLEM IS IN THE FRONT END. IF YOU FEEL A VIBRATION IN THE DRIVERS SEAT THE PROBLEM IS IH THE REAR END. YES THE VIBRATION FROM THE FRONT STEERING AND THE DRIVERS SEAT IS A WAY TO TELL WHERE THE PROBLEM IS

CV joint or replace axcle

This sounds like your CV joint is bad if it is a front wheel drive. The CV joint will click or thump when you are turning. It will also make your front wheels shake if it is realy bad.

The problem is the rear thermostatic expansion valve (TXV).
It is behind the left rear quarter trim panel.You can pull the panel back to access the valve to make sure that that is the noise.If so you will nedd to have AC system evacuated and replace valve and recharge the system. The part number is YF2Z-19849-AA.

Your springs should not be the issue, the body will rot around them before they do. It sounds like shocks to me. I would probably look into all 4 corners, but the rears at least.