my 2006 chevy cobalt with 2.2 l. motor has a engine fan problem. you can start car and fan will be off, but when fan does come on it will not shut off unless you turn the car off.

I can assume the fan is electrical? It it is there is a thermocouple switch connected to the fan switch that will come on when the engine temperature reaches a set level. It will remain on as long as the engine remains above a set temperature. If your dashboard has an engine temperature readout, see how hot it is when the fan comes on. Usually above 180 degrees and shuts off under 160. It may be set for different temperature settings so don't use those numbers as factual. My Corvette engine is built up, and the adage is, the more horse power, the hotter the engine. I replaced my clutch fan with electrical and bypassed the thermocouple switch altogether and run them all the time. Living in Southern California, it is hot 8 months out of the year and we have gridlock traffic a lot, so my car stays in the operating range all the time. Make sure you have the correct amount of coolant and the engine is not running hot for some reason. It can't be checked at highway driving times, but it may kick off then, and kick back on when you slow down or get into heavy traffic when the normal air flow need a fan to keep the engine cooler. A good radiator place should have a temp gun to tell if you engine is overheating when you pull into their garage.