The car turns on the warm air to help the cooling system dissipate heat, which should not cause the fan of the water tank to rotate. The rotation of the fan in the water tank should be caused by the high water temperature.
The radiator fan of an automobile mostly works by electronic temperature control, so generally, the fan will start to rotate normally only when the water temperature in your car reaches the required temperature. If the temperature is too low, the radiator fan will not rotate.