Normal Field Extension $X^4 -4$ has a root in $\Bbb Q(2^{1/2})$ but does not split in $\Bbb Q(2^{1/2})$ implying that $\Bbb Q(2^{1/2})$ is not a normal extension of $\Bbb Q$ according to most definitions. 
But $\Bbb Q(2^{1/2})$ is considered a normal extension of $\Bbb Q$ by everybody. What am I missing here?

You are missing the fact that $x^4-4$ is not irreducible over $\mathbb{Q}$: $x^4-4 = (x^2-2)(x^2+2)$.
The definition you have in mind says that if $K/F$ is algebraic, then $K$ is normal if and only if every irreducible $f(x)\in F[x]$ that has at least one root in $K$ actually splits in $K$. Your test polynomial is not irreducible.
