The vehicle starts over heating if I do anything above 100 on the speed dial also it tends to over heat faster ones the air conditioning is turned on

I have a problem with a Toyota Avensis diesel. In a traffic jam, the temperature needle suddenly raised to the top. When stopping the car, I noticed the fan did not start working, even though it did with the air conditioning. The coolant deposit was empty and so I added around two liters, which were almost gone two days later. I then drove some blocks at a low speed but was not able to determine whether the car overheated or not because the needle remained at its normal position, in the middle. There is nothing wrong in the oil plug and there is no white smoke coming out of the exhaust. I do not even see a coolant stain in the floor. If it were the temperature sensor, the needle would move? Is the sensor related to the coolant consumption? Is it a damaged cylinder head gasket? Thank you.

When it's 8C outside and the air conditioner is set to 23C, if the Automatic button is pressed, the fan is at 0. Manual start and fan speed adjustment work.