Let us consider, for example, the extension $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3}) : \mathbb{Q}$. To check if this extension is normal, would it be sufficient to show that the extension is not the splitting field of
$$p(x) = (x^3 - 2)(x^2 - 3) $$
(because it isn't, $p(x)$ has a factor $(x^2 + \sqrt[3]{2}x + \sqrt[3]{2}^2)$ that has no roots in $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3})$)?
In fact, it's sufficient to check that the polynomial $$q(x) = x^3-2$$ is irreducible over $\mathbb Q$, contains one root in the extension you mentioned, but doesn't split over this extension. Your comments already show this is true.