Electronic factors for the field and mass isotope shifts in the 6p 2P3/2 → 7s 2S1/2 (535 nm), 6p 2P1/2 → 6d 2D3/2 (277 nm), and 6p 2P1/2 → 7s 2S1/2 (378 nm) transitions in neutral thallium were calculated within the high-order relativistic coupled cluster approach. These factors were used to reinterpret previous experimental isotope shift measurements in terms of charge radii of a wide range of Tl isotopes. Good agreement between theoretical and experimental King-plot parameters was found for the 6p 2P3/2 → 7s 2S1/2 and 6p 2P1/2 → 6d 2D3/2 transitions. It was shown that the value of the specific mass shift factor for the 6p 2P3/2 → 7s 2S1/2 transition is not negligible compared with the value of normal mass shift in contrast to what had been suggested previously. Theoretical uncertainties in the mean square charge radii were estimated. They were substantially reduced compared with the previously ascribed ones and amounted to less than 2.6%. The achieved accuracy paves the way for a more reliable comparison of the charge radii trends in the lead region.