The bit in bold is indicative of a knackered rad cap. A towel wrapped round it won't help, not even close.
It is possible that there may be a crack in the rad near the rad cap; to investigate this I'd thoroughly dry the area with a towel, run the engine to temperature and look for where the steam is coming from, and look where the rad is getting wet from the steam. Hopefully that would reveal something. Or, get a garage to do a pressure test on the cooling system: that would find the leak.
If it were me, unless I can see a crack in the radiator, I'd change the rad cap and try it again.