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Novel physical properties not seen in their individual layers are obtained from stacked layers of materials. In this report, using the first-principles calculations, we study the electronic band structures of heterostructures composed of graphitic (graphene-like) bilayers of g-ZnO/ZnX (X = S, Se, and Te) semiconductors. The g-ZnO layer is largely planar, but the ZnX layer is somewhat corrugated. Our calculations reveal on the stable formations of a type-II band alignment in bilayers with an AB stacking order of OZnO-ZnZnX and ZnZnO-XZnX. The optical bandgaps are much lowered as compared to the monolayers of ZnO and ZnX constituents, with the conduction band contributed by g-ZnO and valence band by ZnX. There is a quadratic dependence of the bandgap on the external electric biases applied in perpendicular to the interface. This nonlinearity manifests an superposed effect of an interlayer orbital hybridization and associated charge transferring, along with the coexistence of an electric dipole and a quadrupole. There is an overall charge transfer of valence electrons from the ZnX layer to the g-ZnO layer, leading to a pair of highly charge polarized surfaces along with a large built-in potential. The already lowered bandgaps further change with the external bias. This tunability of Eg for g-ZnO/ZnX bilayer heterostructures holds promises for optoelectronic applications covering a wider spectral range of the solar spectrum.