No: Turing assumes the brain's a machine
Whether or not the brain is a machine is part of what's at issue, because if brains were machines then obviously some machines (brains) would be able to think. So Turing begs the question, by assuming part of what he's trying to prove.
The Apter Argument"Turing is arguing that there need be no difference in principle between the organisation of information processing in the computer and in the human brain. If one was to program a computer to learn for itself how to do something (i.e. program it with a 'child-brain' rather than a an 'adult brain') and if one was to input something like the same amount of information a human receives, both formally and informally, throughout childhood, then we should have a chance of producing a computer which would be able to pass the test he proposes for it. Of course, computers would have to have much larger memory stores and much better information input facilities than they have at present. But Turing regards such problems as essentially engineering problem, and problems which will sooner or later be overcome--his argument is at the more philosophical level of what is and is not in principle possible" (Apter, 1971, p. 68).
"There are obvious objections that can be made to the argument. For one thing it begs the question: if the brain is a machine, then we should be able to get a computer to behave as the brain does. But the objection many people have to the idea of a computer thinking is the belief that the brain is not just a machine. (The opposite argument that defines thinking at the outset as a uniquely human ability also of course begs the question)" (Apter, 1971, p. 68).
Note: A similar argument is put forward in Butter et. al (1995), Chapter 5.
Sources: Apter, Michael (1971)
The Computer Simulation of Behavior. London: Hutchinson and Company, LTD.
Button, Graham (1995)
Computers, Minds and Conduct. Cambridge: Polity Press.