Solar cells based on polycrystalline Cu(In,Ga)Se(2) absorber layers have yielded the highest conversion efficiency among all thin-film technologies, and the use of flexible polymer films as substrates offers several advantages in lowering manufacturing costs. However, given that conversion efficiency is crucial for cost-competitiveness, it is necessary to develop devices on flexible substrates that perform as well as those obtained on rigid substrates. Such comparable performance has not previously been achieved, primarily because polymer films require much lower substrate temperatures during absorber deposition, generally resulting in much lower efficiencies. Here we identify a strong composition gradient in the absorber layer as the main reason for inferior performance and show that, by adjusting it appropriately, very high efficiencies can be obtained. This implies that future manufacturing of highly efficient flexible solar cells could lower the cost of solar electricity and thus become a significant branch of the photovoltaic industry.