It certainly sounds like head gasket failure, or possibly a damaged block causing combustion gases to enter the cooling system.

Get the car up to operating temperature, then take off the header tank cap carefully and watch inside to see if you get bubbles coming through. If there is a steady stream of bubbles then that is almost certainly pointing at gasket/block failure.

Smell inside the header tank. Does it smell of exhaust gases? You could also take it to a mechanic where he will do a sniff test, which tests for exhaust gases being present in the header tank, this will confirm it for you either way.