Transcendental Extension decomposition

Say you have $K \subset L$ transcendental extension. I am wondering if the following is true: if $\exists M$ such that $K \subsetneq M \subsetneq L$ with L algebraic over M, then L is not purely transcendental over K.

If so, then in this question, wouldn't it be enough to show that $y^2 - x^3 + x$ is irreducible over $\mathbb{Q}(x)[y]$? and if so, any help on how to do that?

Counterexample: $$\mathbb{Q} \subseteq \mathbb{Q}(x^2) \subseteq \mathbb{Q}(x)$$