As someone that used to be tasked with clearing other people’s shit at work, here’s how you fix a clogged toilet without a plunger:

first of all DO NOT FLUSH! Stay tf away from the handle on the toilet until the clog is cleared. shutoff the water if possible, there should be a valve between the toilet and the wall that you turn 90°. Even then there’s still water in the tank, so I repeat do not flush!

If the clog isn’t 100% and some water can slowly drain through, leave it for a few minutes and come back. You want the water level in the bowl as low as possible. Next, fill a small trash can with hot water. not boiling, just the hottest you can get from the tap. Now dump that hot water in the toilet bowl. Be quick about it, but not so quick that you can’t stop yourself from overflowing

The hot water and large volume helps break things up. manually pouring you can dump a lot more water in the bowl a lot faster than flushing normally will without the risk of overflowing.