To refute this proposition, you'd need to present evidence of a human being performing an operation that can't be done by a Turing machine.

That's quite easy: I can lift a rock, a Turing machine can't. A Turing machine can only manipulate symbols on a strip of tape, it can't do anything else that's physical.

Your claim that consciousness (whatever we mean when we say that) is possible for Turing machines, rests on the assumption that consciousness is about computation alone, not about computation+some unidentified physical reaction that's absent to pure Turing machines resting in a box on a table.

That consciousness is about computation alone may indeed end up true, but it's as yet unproven.