In order to investigate the origin of multiple populations in globular clusters (GCs), we have constructed new chemical evolution models for proto-GCs where the supernova blast waves undergo blowout without expelling the ambient gas. Chemical enrichments in our models are then dictated by the winds of massive stars together with the asymptotic-giant-branch stars ejecta. We find that the observed Na-O anti-correlation can be reproduced when multiple episodes of starburst and enrichment are allowed to continue in proto-GCs. The “mass budget problem” is mostly resolved by our models without ad-hoc assumptions on star formation efficiency, initial mass function, and significant loss of first-generation stars. Interestingly, ages and chemical abundances predicted by this chemical evolution model are in good agreements with those independently obtained from our stellar evolution model for the horizontal-branch. We also discuss observational evidence for the GC-like multiple populations in the Milky Way bulge.