No description of model but a poor fan thermal coupling might do this on an otherwise good cooling system if it has one. Easily tested. Bring car to operating temperature. Freewheel turning of the fan upon shut down more than a time or so the fluid fan coupling may not be as strong as it should be.

Or if the fan can be kept immobile with a rolled up newspaper on a hot restart. The coupling may again be too weak. Others may suggest better tests. Plus the problem of course may be otherwise. Watch how you preform the last test as it could be dangerous in the wrong hands. Or avoid that test if uncertain.

Waste heat generation at idle is usually less than when under load. Wind created through the rad when underway removes heat. You may be getting away with this again if the cooling system is good. At idle that breeze is missing or of less force unless the fan is functioning as it should.