The lead isotopic chain is a benchmark for nuclear-structure theory. Important information about nuclear properties can be obtained from an analysis of changes in mean-square charge radii. They are determined from experimental isotope-shift values, provided that the corresponding atomic factors have been established. However, there is a marked discrepancy between atomic factors determined using different procedures. The uncertainties of the existing atomic calculations are not well established. We performed high-precision relativistic coupled-cluster calculations of the field-, normal mass-, and specific mass-shift factors for both the 283.3- and 723-nm transitions in neutral lead. For the 723-nm transition, we adopted semiempirical factors obtained by combining the calculated 283.3-nm factors with an experimental King plot. These agree with the direct calculations. Combining the adopted atomic factors with the available isotope-shift values, we determined changes in mean-square charge radii relative to ${}^{208}Pb$ with an estimated atomic-theory uncertainty of about 1%. These results provide a reliable basis for testing nuclear-structure theories.