Turing machine and boolean circuits
The Boolean circuits you are referring to are a non-uniform model. This means that, for every input length, you have a different circuit. When we say that Boolean circuits solve a particular problem what is actually meant is that a family of circuits does, which is an infinite sequence $(C_n)_{n \in \mathbb{N}_0}$, $C_n$ being the circuit for inputs of length $n$.

Moreover, Boolean circuits in general are intrinsically non-uniform models of computation anyway since the input gates count as part of the circuit. If you insist on an arbitrary number of input gates, then you necessarily end up with a circuit of infinite size (which is a no-no). (In fact, this is why we use the notion of constructability to obtain uniform circuits; any sort of uniformity must be extrinsic to the circuit.)