  We prove that the number of quartic fields $K$ with discriminant
$|\Delta_K|\leq X$ whose Galois closure is $D_4$ equals
$CX+O(X^{5/8+\varepsilon})$, improving the error term in a well-known result of
Cohen, Diaz y Diaz, and Olivier. We prove an analogous result for counting
quartic dihedral extensions over an arbitrary base field.
