A complex manifold is called an Oka manifold if every holomorphic map from a convex set in a Euclidean space to the manifold is a limit of entire maps. We show that every Oka manifold admits families of noncompact complex curves of any given topological type, endowed with a prescribed family of conformal structures, with approximation on compact Runge subsets. A similar result holds for families of minimal surfaces and null holomorphic curves in Euclidean spaces.