Antibiotic resistance is a worldwide challenge. A potential approach to block resistance is to simultaneously inhibit WT and known escape variants of the target bacterial protein. Here, we applied an integrated computational and experimental approach to discover compounds that inhibit both WT and trimethoprim (TMP) resistant mutants of E. coli dihydrofolate reductase (DHFR). We identified a novel compound (CD15-3) that inhibits WT DHFR and its TMP resistant variants L28R, P21L and A26T with IC50 50– 75 \textmu M against WT and TMP-resistant strains. Resistance to CD15-3 was dramatically delayed compared to TMP in in vitro evolution. Whole genome sequencing of CD15-3-resistant strains showed no mutations in the target folA locus. Rather, gene duplication of several efflux pumps gave rise to weak (about twofold increase in IC50) resistance against CD15-3. Altogether, our results demonstrate the promise of strategy to develop evolution drugs - compounds which constrain evolutionary escape routes in pathogens.