We report analytic investigation of the electronic properties of heterostructures comprised by films of three-dimensional topological insulator (TI) and normal insulator (NI) revealing strong interface and size effects on their spin transport characteristics. Imposing at the TI/NI interfaces the natural boundary conditions within a continual scheme, we show that the intrinsic spin Hall response of the NI/TI/NI trilayer can be controlled by tuning the interface potential (IP) together with the TI film thickness. We predict a series of interface induced quantum transitions between topological insulating phase and trivial band insulator phase. We calculate the phase diagram of the NI/TI/NI trilayer and establish an appropriate range of the IP strength and the TI film thickness to realize the regime of quantum spin Hall effect. Our results can provide a useful guide in choosing the relevant material parameters to facilitate the detection of quantum spin Hall conductivity in the TI/NI heterostructures.