Unless you have the ignition/engine interlocked with a turbo timer, nothing stays alive, as mentioned above, when the key is removed.
When the engine is left running with a timer, the fan may come on if the engine is in the correct temp range (read - higher ranger), but won't otherwise.
If you wire up the fan circuits to a switch/relay setup, then and only then can you control the operation of the fan other than designed by the factory.
The engine heat soaks when it is shut off, and this is the effect that you see when the temp gauge is so high upon the next key forward cycle/engine start.