I have bought a used Asus G53J and every time I turn it on, it shuts off after 5 or 10 minutes, I don't have the original adapter and I'm working with a Toshiba adapter. Is the Adapter the source of the problem or is it something with the system?
About your problems, is the battery charging ?
You could have a hardware problem (overheating )
Got to asus website and get the voltage, amps and wattage output of the original psu. If the specs are similar, it should work fine. If not, it might be the cause of the problem.
Also you might want to boot in safe mode with networking and try to install coretemp, hwmonitor to look at the temperatures.
Also scan for viruses with malwarebytes rogue killer and adwcleaner if the computer doesn't crash