Size, depth, and time of circuits
No.  Every function can be implemented with a polynomial-depth circuit, but not all function families can be implemented with a polynomial-time algorithm.

In particular, if we're allowed unbounded fan-in/fan-out, you can implement every function in depth 2 using a CNF or DNF formula for the function (but the size of the circuit is exponential).  If we're only allowed constant fan-in/fan-out, you can simulate that with linear depth (use a $n$-layer tree to make exponentially many copies of each input; then 1 layer to evaluate the CNF clauses; then a $n$-layer AND tree).