Many cellular populations are tightly packed, such as microbial colonies and biofilms, or tissues and tumours in multicellular organisms. The movement of one cell in these crowded assemblages requires motion of others, so that cell displacements are correlated over many cell diameters. Whenever movement is important for survival or growth, these correlated rearrangements could couple the evolutionary fate of different lineages. However, little is known about the interplay between mechanical forces and evolution in dense cellular populations. Here, by tracking slower-growing clones at the expanding edge of yeast colonies, we show that the collective motion of cells prevents costly mutations from being weeded out rapidly. Joint pushing by neighbouring cells generates correlated movements that suppress the differential displacements required for selection to act. This mechanical screening of fitness differences allows slower-growing mutants to leave more descendants than expected under non-mechanical models, thereby increasing their chance for evolutionary rescue. Our work suggests that, in crowded populations, cells cooperate with surrounding neighbours through inevitable mechanical interactions. This effect has to be considered when predicting evolutionary outcomes, such as the emergence of drug resistance or cancer evolution.