Some cars will allow the radio to receive power when the ignition switch is turned to off for about 10 minutes or until a door is opened. The problem may be in this circuit if this is the way the car is set up. Check your owner’s manual.
Your radio should have a means to turn it off while the car is running. Some people, including me, don’t like having the radio on while they drive. I use the radio rather sparingly–there isn’t much on the radio that appeals to me. If you can’t turn the radio off with the switch on the radio, then there really is a problem.
I think you need to take your car to a shop that specializes in electrical work.