We know that if $X$ is a smooth complex projective variety and we assume that there is a dominant morphism $f : X \to Y$ with $Y$ and the general fibers of $f$ rationally connected. Then $X$ itself is rationally connected. (This is established as a corollary of Graber, Harris and Starr’s theorem).

My question is why this fails for rational varieties of higher dimension. I saw in some works about rationally connected varieties that "there exists" some examples of $\mathbb{P}^1$-fibrations over $\mathbb{P}^2$ that are not rational, but I couldn't find these examples explicitely.

Can you give me an idea or references, please? Thank you very much in advance!