My definition of a Turing machine? I’m not sure you know what Turing machines are. It’s a general purpose computer, described in principle. And, in principle, a computer can only carry out one task at a time. Modern computers are fast, they may have several CPUs stitched together and operating in tandem, but they are still fundamentally limited by this. Bodies don’t work like that. Every part of them is constantly reacting to it’s environment and it’s neighboring cells - concurrently.
You are essentially saying, “Well, the hardware of the human body is very complex, and this software is(n’t quite as) complex; so the same sort of phenomenon must be taking place.” That’s absurd. You’re making a lopsided comparison between two very different physical systems. Why should the machine we built for doing sums just so happen to reproduce a phenomena we still don’t fully understand?
I agree that there are similarities in how groups of nerve cells process information and how neural networks are trained, but I’m hesitant to say that’s a whole picture of the human mind. Modern anesthesiology suggests microtubuals, structures within cells, also play a function in cognition.