There's nothing to indicate the human brain can't be implemented in a Turing machine too.
That is not true, this is a research branch from computer science and math, it is called Computability theory, it deals with the limits of expressiveness for different types of theoretical machines and expressions, and the most expressive of all is the Turing machine, and a Turing machine cannot do some stuff, the classic example is the Halting problem, a computer cannot definitely say if an algorithm ever stops (mathematica proven that it cannot do it), but a human can do so quite easily.
One may think that maybe a Turing Machine cannot do something but can simulate another machine that does, but that is also proven to be impossible, it cannot simulate something more expressive than itself.