I have a 2002 Chrysler PT Cruiser non turbo

Leaking head gasket? Have the "sniffer" test done. Your mechanic will draw air from the radiator through a glass cylinder with two chambers partially-filled with a special dark blue liquid. It will turn bright yellow if combustion gases are present. A leaking head gasket commonly causes bubbling in the reservoir before the engine has even warmed up. The radiator hoses won't be hot yet either. Air from the leak can pool under the thermostat causing it to remain closed. They must be hit with hot liquid to open.

Check the cooling fins on the radiator to see if they're corroded and crumbling. The clue is running the heater on high will bring the temperature down.

Check the timing belt for wear and proper tension to be sure the water pump pulley isn't slipping. That would be uncommon but not something to overlook.