Periodic stacking of topologically trivial and non-trivial layers with opposite symmetry of the valence and conduction bands induces topological interface states that, in the strong coupling limit, hybridize both across the topological and normal insulator layers. Using band structure engineering, such superlattices can be effectively realized using the IV-VI lead tin chalcogenides. This leads to emergent minibands with a tunable topology as demonstrated both by theory and experiments. The topological minibands are proven by magneto-optical spectroscopy, revealing Landau level transitions both at the center and edges of the artificial superlattice mini Brillouin zone. Their topological character is identified by the topological phase transitions within the minibands observed as a function of temperature. The critical temperature of this transition as well as the miniband gap and miniband width can be precisely controlled by the layer thicknesses and compositions. This witnesses the generation of a new fully tunable quasi-3D topological state that provides a template for realization of magnetic Weyl semimetals and other strongly interacting topological phases.