Find a counterexample to the statement "If $K\subset M\subset L$ are fields and $L$ is normal over $K$, then $M$ normal over $K$"

Question

I want to find a counter-example of three fields $K$, $M$, and $L$ where the following statement does not hold:

If $K \subset M \subset L$ and $L$ is normal over $K$, then $M$ is normal over $K$.

I started with $K=\mathbb{Q}$ and think it could work with $M=\mathbb{Q}(A)$ and $L=\mathbb{Q}(A,B)$ where $A$ and $B$ are some radicals. I haven't really gotten anywhere with this.