Some conic bundles that are not birationally trivial do the job. For explicit examples, see

Matsuki, Kenji, Introduction to the Mori program, Universitext. New York, NY: Springer (ISBN 0-387-98465-8/hbk). xxiii, 478 p. (2002). ZBL0988.14007,

pages 143-148.

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

K. Matsuki: Introduction to the Mori program, Universitext. New York, NY: Springer (ISBN 0-387-98465-8/hbk). xxiii, 478 p. (2002). ZBL0988.14007,

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.