The evolution of human influenza virus haemagglutinin (HA) involves simultaneous selection to acquire antigenic mutations that escape population immunity while preserving protein function and stability. Epistasis shapes this evolution, as an antigenic mutation that is deleterious in one genetic background may become tolerated in another. However, the extent to which epistasis can alleviate pleiotropic conflicts between immune escape and protein function/stability is unclear. Here we measure how all amino acid mutations in the HA of a recent human H3N2 influenza strain affect its cell entry function, acid stability and neutralization by human serum antibodies. We find that epistasis has entrenched certain mutations so that reverting to the ancestral amino acid identity in earlier strains is no longer tolerated. Epistasis has also enabled the emergence of antigenic mutations that were detrimental to the cell entry function of HA in earlier strains. However, epistasis appears insufficient to overcome the pleiotropic costs of antigenic mutations that impair the stability of HA, explaining why some mutations that strongly escape human antibodies never fix in nature. Our results refine our understanding of the mutational constraints that shape recent H3N2 influenza evolution: epistasis can enable antigenic change, but pleiotropic effects can restrict its trajectory.