Some conic bundles that are not birationally trivial do the job. For explicit examples, see pages 143-148 of

<cite authors="Matsuki, Kenji">K. Matsuki: *Introduction to the Mori program*, Universitext. New York, NY: Springer (ISBN 0-387-98465-8/hbk). xxiii, 478&nbsp;p. (2002). [ZBL0988.14007](https://zbmath.org/?q=an:0988.14007),</cite> 

The case of a cubic threefold $W_3 \subset \mathbb{P}^4$, cited by Roy Smith in his comment, belongs to this family of counterexamples. In fact, the blow-up $X=\mathrm{Bl}_L(W_3)$ of $W_3$ along a line $L \subset W_3$ is a conic bundle over $\mathbb{P}^2$. By Clemens-Griffths we know that $W_3$ is not rational, so $X$ is not rational, and this implies that its conic bundle $X \to \mathbb{P}^2$ is not birationally trivial.