It is an interesting problem to find asymptotic expressions for the number of degree n number fields with specified Galois closure G and bounded discriminant. We prove that the number of quartic fields K with discriminant < X whose Galois closure is \(D_4\) equals \(CX+O(X^{5/8+\epsilon})\), improving the error term in a well-known result of Cohen, Diaz y Diaz, and Olivier. In order to carry this out, we establish a result for counting relative quadratic extensions with a power saving error term for which the implicit constant only depends on the degree of the base field. This is joint work with Amanda Tucker.