Would anyone know why my computer screen and tower keep shutting down while I'm online?

There can be several root causes.

1: Overheating.

Your PC may have accumulated an overly large amount of dust over the years and may be heating up your processor to 80 celsius or above, and therefore triggering an automatic safety shutoff mechanism.

2: Power Supply Unit Death

Your PSU or Power Supply Unit - is a large box within the computer that converts outside voltage into internal smaller voltages for the computer to handle for certain components. Before the PSU dies, the computer will often display such symptoms as random shut offs.

3: Unclean Power

The power circuitry within your home may be faulty and may not take the full voltage of your computer along with several high voltage household appliances such as washing machines, air conditioners, and refridgerators. If you have your computer connected to a busy circuit, you may be causing your fuses to bust, and then restart the circuit.

I'd recommend checking steps from 1 to 3 for most likely causes. Dust is probably the culprit in this one, and if not the dust then the power supply - and if that turns out fine, then it's your power.

take the cover off the tower and turn it on and look at the fan on the CPU on th motherboard. If it is not working or running really slow your puter is overheating and shutting down to protect the cpu. replace the fan. Its about 14 or 15.00 at the computer shop and easy to take out and install.

you have a virus, I think possibly the 60 second one. After 60 seconds of connecting online it will switch of your system, I had it before but I went to my local pc hardware store and they gave me a restoration disk