We have measured the hyperfine splitting of the 7P(1/2) state at the 100 ppm level in Fr isotopes (Fr-206g,Fr-206m,Fr-207,Fr-209,Fr-213,Fr-221) near the closed neutron shell (N = 126 in Fr-213). The measurements in five isotopes and a nuclear isomeric state of francium, combined with previous determinations of the 7S(1/2) splittings, reveal the spatial distribution of the nuclear magnetization, i.e., the Bohr-Weisskopf effect. We compare our results with a simple shell model consisting of unpaired single valence nucleons orbiting a spherical nucleus, and find good agreement over a range of neutron-deficient isotopes (Fr207-213). Also, we find near-constant proton anomalies for several even-N isotopes. This identifies a set of Fr isotopes whose nuclear structure can be understood well enough for the extraction of weak interaction parameters from parity nonconservation studies.