It does sound like it is overheating, but you could try just powering the laptop with the battery out... Then maybe checking the heat vent on the side of the laptop to see if it feels hot; check if the heat vent is not clogged up with dust, hair etc... Also when you use the laptop make sure it is on a flat surface so that the CPU fan can bring in cold air from around the laptop. Lastly, make sure you can hear the fan spinning.
Questions:
How long does it take before the laptop with turn off?