2004 Toyota Camry with a faulty turn signal switch-what could be causing this?

Did anyone change/replace any Tail/Brake/Parking light bulbs?

No but before this problem occurred we had the starter replaced. It only happens when driving, turn the signal light on and when the brakes are pressed down the flickering starts and the radio, wind shield wipers, and power windows stop working. I purchased a flasher relay but uncertain where it goes on the car.

Now, allow me to clarify this issue. Your turn signals operate normally until you step on the brake pedal of this vehicle. When the brake pedal is pressed both the left and right turn signal indicators on the dash flicker when you move the turn signal lever for a left or right turn. Am I correct?

Yes. Correct. That was the problem but now I have purchased a new flasher relay and changed that out. Now my only issue is while driving the car all lights, radio, a/c-heat, windows, windshield wipers, and dashboard lights go out all at once. The speedometer stops working as well for like 2-3 mins and then it blink back on. Does this sound like a computer problem or a bad battery? Note: car still drives, no skips or funny sounds from the running car or motor. Just inside the car the entire dashboard goes out.

Sounds like a bad connection.....maybe a bad ground. This is an instance when one would consult a wiring diagram for the Cluster circuits. Also....there might be a "Service Bulletin" addressing this issue. Try this....while stopped with the engine idling....smack the top of the dash with the palm of your hand. Then...drive the vehicle and again smack the dash. If the problem occurs ...you probably have a bad connection at the rear of dash. Also...do a Google search for this make/model/year+++"Service Bulletins" Some sites will request a FEE to read the SB. Not necessary. Just get the SB#. Get back to us for further diagnosis.