If thermostat is new, I would replace radiator, non distilled water and years of scale buildup causes inefficency. Either that or the water pump is almost done. Cap not holding pressure is good to check for.

I would drain the system and flush out the radiator and the engine block. There is probably a blockage in the radiator. When you put tap water in the system it will leave hard water deposits like calcium that can block coolant flow. This is why distilled water is recommended because it is the purest form of water. The radiator may need to be professionally flushed at a shop. You can try yourself by using a garden hose. Keep me posted and I will answer any other questions for you. Please rate me on this answer.