A (k,g)-cage is a k-regular graph with girth g that has the fewest number of vertices. It has been conjectured (Fu et al., J Graph Theory 24 (1997), 187–191) that all (k,g)-cages are k-connected for k ≥ 3. A connected graph G is said to be superconnected if every minimum cut-set S is the neighborhood of a vertex of minimum degree. Moreover, if G − S has precisely two components, then G is called tightly superconnected. It was shown (Xu et al., Ars Combin 64 (2002), 181–192) that every (4,g)-cage is 4-connected. In this article, we prove that every (4,g)-cage is tightly superconnected when g is even and g ≥ 12.