Picking up where I left off in the comments: if $f$ is irreducible and has a repeated root in its splitting field, then $f' = 0$. But if we write $f(x) = \sum a_nx^n$, then $0 = f'(x) = \sum na_nx^{n-1}$ means that whenever $a_n \neq 0$ we have $n = 0$ in $K$, which is to say $p | n$, where $p$ is the characteristic of $K$. This says precisely that $f(x) = g(x^p)$ for some polynomial $g$.
For your other question: if $M/K$ is normal then $M/L$ is normal automatically, but $K/L$ need not be. For example, take $M = \mathbb{Q}[2^{1/4},i]$, which is Galois (in particular, normal) with group $D_4$, the dihedral group of the square. The subextension $L = \mathbb{Q}[2^{1/4}]$ is not normal: there is an automorphism of $M$ which sends $2^{1/4} \mapsto 2^{1/4}i \notin L$. Alternatively, one can show that $L$ corresponds via Galois theory to a subgroup in $D_4$ generated by a reflection, which is not normal.