I know that I am looking for a counterexample, the statement is not true when the Disc(K) and the Disc(L) are not coprime.
I have been trying to use $Disc(K)=[\mathcal{O}_K:K]^2Disc(\mathcal{O}_K)$, but am not having much success.
I have particularly been looking at the two fields generated by $f(x)=x^2-3$ and $g(x)=x^2-2$.
Any help would be greatly appreciated I have been trying to do this for ages.