AI is most definitely an algorithm. It runs on a computer, what else could it be? Humans didn't create the algorithm directly, but it certainly exists within the machine. The computer takes an input, does a series of computing operations on it, and spits out a result. That is an algorithm.
As for humans, there is no way you can look at the behavior of a human and know for certain it is not a Turing machine. With a large enough machine, you could simulate any behavior you want, even behavior that would look, on first observation, to not be coming from a Turing machine; this is a form of the halting problem. Any observation you make that makes you believe it is NOT coming from a Turing machine could be programmed to be the output of the Turing machine.
> With a large enough machine, you could simulate any behavior you want
This is not exactly true, depending on what you mean by behavior. There are mathematical functions we know for a fact are not computable by a Turing machine, no matter how large. So a system that "behaves" like those functions couldn't be simulated by a TM. However, it's unclear whether such a system actually could exist in physical reality - which gets right back to the discussion of whether thinking is beyond Turing completeness or not.
As for humans, there is no way you can look at the behavior of a human and know for certain it is not a Turing machine. With a large enough machine, you could simulate any behavior you want, even behavior that would look, on first observation, to not be coming from a Turing machine; this is a form of the halting problem. Any observation you make that makes you believe it is NOT coming from a Turing machine could be programmed to be the output of the Turing machine.