Photoelectron spectroscopy, optical characterization, and density functional calculations of ZnO1-xSx reveal that the valence-band (VB) offset E(v)(x) increases strongly for small S content, whereas the conduction-band edge E(c)(x) increases only weakly. This is explained as the formation of local ZnS-like bonds in the ZnO host, which mainly affects the VB edge and thereby narrows the energy gap: E(g)(x=0.28) approximately E(g)(ZnO)-0.6 eV. The low-energy absorption tail is a direct Gamma(v)-->Gamma(c) transition from ZnS-like VB. The VB bowing can be utilized to enhance p-type N(O) doping with lower formation energy DeltaH(f) and shallower acceptor state in the ZnO-like alloys.