We extract the proton magnetic radius from the high-precision electron-proton elastic scattering cross section data. Our theoretical framework combines dispersion analysis and chiral effective field theory and implements the dynamics governing the shape of the low-$Q^2$ form factors. It allows us to use data up to $Q^2sim$ 0.5 GeV$^2$ for constraining the radii and overcomes the difficulties of empirical fits and $Q^2 rightarrow 0$ extrapolation. We obtain a magnetic radius $r_M^p$ = 0.850 $pm$0.001 (fit 68%) $pm$0.010 (theory full range) fm, significantly different from earlier results obtained from the same data, and close to the extracted electric radius $r_E^p$ = 0.842 $pm$0.002 (fit) $pm$0.010 (theory) fm.
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