Programs to circuit conversion
The Cook&ndash;Levin theorem shows how to construct a circuit of size $f(n)^{O(1)}$. I'm not sure what's the best exponent.

The opposite direction is impossible, since circuits of size 1 can compute the following language:
$$ \{ w : \text{the $|w|$th Turing machine halts on the empty input} \}. $$
More generally, circuits of size 1 can compute any language in which the answer depends only on the length of the input.