Certainly every purely inseparable simple extension must be normal, since the minimal polynomial of the generating element $a$ must look like $X^{p^r}-a$, which splits in the extension, so the example can't be simple.
Other than that I'm pretty out of ideas though.
As you can see from my remark above, I have never thought of this question in any depth. I’m troubled by what I write below, but let’s look:
A purely inseparable extension $K\supset k$ has separable degree $1$, in other words, there is precisely one embedding $K\hookrightarrow k^{\text{ac}}$ that is identity on $k$. Since there is only one such map, it necessarily carries $K$ into itself, so that $K\supset k$ will be normal.