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

Question

Let $K\subset L\subset M$ be a tower of finite extensions. If $K\subset L$ and $L\subset M$ are normal, then $K\subset M$ is normal. Is the statement true? If yes, give a proof and if not, give a counterexample. Any hints? I still don't know whether it is true or not.