Carbon storage in forests is a cornerstone of policy-making to prevent global warming from exceeding 1.5°C. However, the global impact of management (for example, harvesting) on the carbon budget of forests remains poorly quantified. We integrated global maps of forest biomass and management with machine learning to show that by removing human intervention, under current climatic conditions and carbon dioxide (CO2) concentration, existing global forests could increase their aboveground biomass by up to 44.1 (error range: 21.0 to 63.0) petagrams of carbon. This is an increase of 15 to 16% over current levels, equating to about 4 years of current anthropogenic CO2 emissions. Therefore, without strong reductions in emissions, this strategy holds low mitigation potential, and the forest sink should be preserved to offset residual carbon emissions rather than to compensate for present emissions levels.