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We study the moduli spaces of elliptic K3 surfaces of Picard number at least 3, that is, U⊕⟨−2k⟩‐polarized K3 surfaces. Such moduli spaces are proved to be of general type for k⩾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. [ABSTRACT FROM AUTHOR]