This paper is devoted to the important yet unexplored subject of crowding effects on market impact, that we call "co-impact". Our analysis is based on a large database of metaorders by institutional investors in the U.S. equity market. We find that the market chiefly reacts to the net order flow of ongoing metaorders, without individually distinguishing them. The joint co-impact of multiple contemporaneous metaorders depends on the total number of metaorders and their mutual sign correlation. Using a simple heuristic model calibrated on data, we reproduce very well the different regimes of the empirical market impact curves as a function of volume fraction $\phi$: square-root for large $\phi$, linear for intermediate $\phi$, and a finite intercept $I_0$ when $\phi \to 0$. The value of $I_0$ grows with the sign correlation coefficient. Our study sheds light on an apparent paradox: How can a non-linear impact law survive in the presence of a large number of simultaneously executed metaorders?
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