The only serious counter-argument I've heard against the hypothesis that the brain goes further than a Turing machine is that the brain does not implement an algorithm, with one input, some processing steps, and one output. Instead, it processes information as it comes, this information comes during the processing of previous information and the output partly loops back to the input. Short-circuits everywhere, no clo…
Computers do this all the time. Simplest example is "cancel" button everywhere.