The flaw in your argument is that "surpass" doesn't just mean that you should be able to run all algorithms, it includes a notion of complexity, i.e. how many time steps you will take to simulate an algorithm.
How do you simulate an algorithm with a Turing machine? A Turing machine consists of a finite state machine and an infinite tape. A Turing Machine does run an algorithm, determined by its initial state and the state transition matrix, but what I think you are talking about is Universal Turing Machines (UTM) that can read "code" (which is usually a description of another Turing machine) written on a "code segment" of the tape and then simulate that machine on input data written on the "data segment" of the tape.
Turing machines can differ in the number of states in their finite state machines (and also in the alphabet they write on the tape but any finite alphabet is easily encoded in binary so this should not be the big reason for differences among Turing machines). So, you can have UTMs with bigger state machines and UTMs with smaller state machines. The bigger UTM could possibly surpass the smaller one if they use the same encoding for the "code" part of the tape.
You can also play around with the code used to describe the TM being simulated. This code could be C++, for example, or could be a Neural network with the synapse strength written down as a matrix. Which description is better for computation depends on the problem.
An example comparison among UTMs with different state machines: consider different compilers for the same language, say C++. Both of them will first compile C++ to assembly and then run another UTM which reads and executes assembly (your physical CPU). So, a better compiler will run the same code faster.
Back to humans vs computers, humans are neural networks that run algorithms like those you would write in C++. This involves a costly and inefficient conversion of the algorithm into hand movements. A computer uses a compiler to convert C++ to assembly that it can run natively, so it's able to do a much more efficient implementation of C++ code. Alternately, humans have a ton of neurons, and the neural code, i.e. synapse strength, is hard to read, so current computers cannot run that code yet.