How to relate circuit size to the running time of Turing machine
The way to show that circuits are equivalent to TMs is as follows: for a TM, you encode it in binary (i.e states, transitions, etc.)

Then, you construct a circuit whose gates represent the transitions between each configuration of the TM. Since configuration transitions are local, then you only need local gates (i.e gates that depend on a constant number of inputs). 

The idea is that you can compute configuration $c_i$ from configuration $c_{i-1}$ by examining the contents of 3 adjacent cells, and if the head of the machine is there, update them accordingly. It's rather technical to write formally, but it's a generally simple proof (See "Computational complexity" by Arora and Barak for a proof).

Using this proof, you see that the size of the circuit is polynomial in the running time of the machine and the input length, because the circuit is constructed in "levels", where each level corresponds to a configuration of the TM.