Are all $\mathbf{P^1}$-fibrations projective bundles?

Question

Maybe there is a very simple answer to my question or a counterexample - I don't see it:
Let $\mathcal{X}\rightarrow C$ be a flat, smooth family with fibres $\cong \mathbf{P^1}$ of dim $2$ over a smooth projective base $C$ (maybe just $\mathbf{P}^1 $).
Is then necessarily $\mathcal{X}$ isomorphic to $\mathbf{P}(\mathcal{E})$ for a rank $2$-bundle on $C$? The only fibrations I know with these properties are the Hirzebruch surfaces $\mathbf{F}_n=\mathbf{P}(\mathcal{O}\oplus\mathcal{O}(n))$, who are obviously projective bundles, so is this true more generally? Thank you