ZnO/Mg0.22 Zn0.78 O quantum wells were studied by excitation-intensity-dependent luminescence at 10 K. The samples were grown by laser molecular-beam epitaxy on ScAlMgO4 substrates to evaluate the well width dependence (1 to 10 nm) of exciton transition energies. Under weak excitation, the photoluminescence shows a quantum-confined Stark effect for the wide wells. The well width dependences of the experimental transition energies are compared with previously reported calculations to evaluate the electric field due to spontaneous and piezoelectric polarizations. The internal electric field is comparable with 650 kV/cm. With an increase in excitation intensity, blueshift of the luminescence was observed, suggesting photoexcitation screening of electric fields.