Abstract Forests provide many ecosystem services that strongly depend on species diversity, as illustrated by the repeatedly observed diversity–productivity relationships (DPRs). These forest DPRs are assumed to result mostly from complementarity between species at the tree level whilst emerging community‐level processes remain poorly explored. In this study, we propose that the ‘tree packing effect’ (TPE), where species diversity promotes productivity by positively impacting maximum stand density, is an important determinant of DPRs. We tested the two components of TPE: (i) whether maximum stand density increases with species richness and (ii) whether this higher stand density allowed by species richness promotes forest productivity. First, relying on national forest inventories of six European countries (NFIs, totaling 2,367,776 trees), we fitted self‐thinning lines to examine whether these lines were influenced by plot species richness. We showed that maximum stand density increases with tree species richness in Europe, in all but one country. This trend was notably stronger in extreme climates. Second, we ran a large simulation‐based experiment (including 7,024,815 simulations) with an individual‐based forest dynamics model able to control for stand‐density effects, to quantify DPRs for more than 1000 sites in Europe. Relying on an original method to quantify DPRs at the site level, we compared the strength of DPRs simulated with and without control for stand density. We found positive DPRs up to 10‐times stronger when TPE is at play than when stand density is controlled. This positive effect of diversity on forest productivity through tree packing is also stronger in extreme climates, especially in warm and dry conditions. Synthesis. Highlighting the generality of the TPE in European forests, our results reveal that the effect of diversity on forest functioning is partly mediated by diversity‐driven changes in stand density. This mechanism has been long overlooked in biodiversity—ecosystem functioning studies, but our findings strongly call for its reconsideration, especially in natural forests. It also opens key perspectives for management and climate change mitigation programmes.