So let's say $\deg f \geq 4$, then I'd have at least two common zeroes. Does $C$ have bad reduction at $3$ then?

Let's just work over $\mathbb Q$, for simplicity. Then a naive model of $C$ is given by the equation above. Then the special fiber (over $\mathbb F_3$) is singular by the Jacobian criterion. But I don't get to the point where $C$ has bad reduction at $p=3$.

@FrancescoPolizzi: You may also assume $\deg(f)\geq 4$ to avoid trivial cases