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In this article we show a $C^{0,α}$-partial regularity result for solutions of a certain class of cross-diffusion systems with entropy structure. Under slightly more stringent conditions on the system, we are able to obtain a $C^{1,α}$-partial regularity result. Amongst others, our results yield the partial $C^{1,α}$-regularity of weak solutions of the Maxwell-Stefan system, as well as the partial $C^{1,α}$-regularity of bounded weak solutions of the Shigesada-Kawasaki-Teramoto model. The classical partial regularity theory for nonlinear parabolic systems as developed by Giaquinta and Struwe in the 80s proceeds by Campanato iteration which relies on energy methods. Our analysis here centers around the insight that, in the Campanato iteration strategy, we can replace the use of energy estimates by "entropy dissipation inequalities" and the use of the squared $L^2$-distance to measure the distance between functions by the use of the "relative entropy". In order for our strategy to work, it is necessary to regularize the entropy structure of the cross-diffusion system, thereby introducing a new technical tool, which we call the "glued entropy".