This could be down to a number of things, but a lot of mechanics would begin by looking at the engine’s fuel supply (pressure and flow) as well as the condition of things like the fuel filter (which can slow flow if it’s dirty or clogged).

It’s also entirely possible that a problem like this is being caused by a faulty temperature sensor which is not telling the engine’s computer the truth about the current temperature. In colder weather and cold starts, an engine needs a richer fuel:air mixture (more fuel, less air) to run smoothly. If the sensor isn’t accurately conveying the temperature to the computer, the mixture could be way off. Poor starting and rough running is definitely a possibility at that point.

An electronic scan of the car should be your first move as this may pinpoint the problem straight away, rather than spending hours of checking things that aren’t the problem.

This is actually quite normal and nothing to be concerned about. If the engine temperature is high enough, the computer will continue to run the cooling fan even though the ignition has been turned off. This helps keep the temperature from spiking when the vehicle stops and is turned off.

Even though this only really lowers the temperature of the coolant in the radiator (not in the actual engine) it also helps to maintain a constant, safe temperature if you restart the car soon after shutting it down.

That said, the fan should only continue running for a couple of minutes at most after you’ve turned the car off. Any longer and you might have a problem with the circuits that power and control the electric fan.

A common cause of overheating can be a failed head gasket which allows the coolant to be consumed by the engine. When this happens, soon there’s not enough coolant remaining to cool the engine and the car overheats.

But it may not be as serious as that. You could simply have a stuck thermostat. This is the component that prevents coolant flowing around the engine when it’s first started, forcing the engine to warm up faster. Then, as the engine gets warmer, the thermostat starts to open to allow the coolant to flow through the engine and do its job, maintaining the correct running temperature in the process.

Other possible causes for overheating include incorrect ignition timing, a slipping fan belt, non-functioning cooling fans, a collapsed radiator hose, blocked radiator, a worn water pump and many others. A good mechanic should be able to make a diagnosis and tackle the problem from there.