Nobody claims that Turing machines in particular are well-suited to the task. What we claim is something more open-ended: Turing-complete languages can simulate each other given a universal encoding (a program we typically call an "emulator") and the simulation is only polynomially slower and larger.

To the details: We don't want just one tape on the Turing machine. It's fine but slow, like using unary Church numerals instead of binary Cantor numerals. We usually assume at least two tapes. Similarly, we often ignore tapes and use a random-access memory instead.

Edit: I guess that this is how we're talking today? From the article:

> It is common to summarize CCTM through the slogan “the mind is a Turing machine”. This slogan is also somewhat misleading, because no one regards Turing’s precise formalism as a plausible model of mental activity. The formalism seems too restrictive in several ways.

The article then lists senses, finite memory, concurrency, and determinism as four ways in which the brain might differ from idealized Turing machines.