  We study the moduli spaces of elliptic K3 surfaces of Picard number at least
3, i.e. $U\oplus \langle -2k \rangle$-polarized K3 surfaces. Such moduli spaces
are proved to be of general type for $k\geq 220$. The proof relies on the
low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran.
Furthermore, explicit geometric constructions of some elliptic K3 surfaces lead
to the unirationality of these moduli spaces for $k < 11$ and for 19 other
isolated values up to $k=64$.
