I found the following statement on the Wikipedia page for normal extensions:
Let L be an extension of a field K. Then:
If E and F are normal extensions of K contained in L, then the compositum EF is also a normal extension of K.
This question has previously been answered here, but the answer made use of the characterisation of normal extensions that involves automorphisms on EF.
I'd like to prove the statement directly from an equivalent definition of normal extensions (stated below), but I've only been successful for the case where E and F are also finite extensions of K.
L is a normal extension of K if every irreducible polynomial in K[X] that has a root in L actually splits over L.
You said you've been successful in the case that $E$ and $F$ are finite, but the general case follows immediately from that case. If $x\in EF$, then $x$ is in the subfield of $L$ generated by finitely many elements of $E$ and $F$. Let $E_0\subseteq E$ be the splitting field of the minimal polynomials of those finitely many elements of $E$ and similarly define $F_0\subseteq F$. Then $E_0$ and $F_0$ are finite and normal over $K$, and $x\in E_0F_0$. But then $E_0F_0$ is normal over $K$, so the minimal polynomial of $x$ over $K$ splits over $E_0F_0$. The minimal polynomial of $x$ over $K$ thus splits over $EF$. Since $x\in EF$ was arbitrary, $EF$ is normal over $K$.