  Electronic factors for the field and mass isotope shifts in the $6p\
^{2}P_{3/2} \to 7s\ ^{2}S_{1/2}$ (535 nm), $6p\ ^{2}P_{1/2} \to 6d\
^{2}D_{3/2}$ (277 nm) and $6p\ ^{2}P_{1/2} \to 7s\ ^{2}S_{1/2}$ (378~nm)
transitions in the 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 $6p\ ^{2}P_{3/2} \to 7s\ ^{2}S_{1/2}$
and $6p\ ^{2}P_{1/2} \to 6d\ ^{2}D_{3/2}$ transitions. It was shown that the
value of the specific mass shift factor for the $6p\ ^{2}P_{3/2} \to 7s\
^{2}S_{1/2}$ transition is not negligible compared to the value of normal mass
shift in contrary to what had been suggested previously. Theoretical
uncertainties in mean square charge radii were estimated. They were
substantially reduced compared to the previously ascribed ones and amounted to
less than 2.6%. The achieved accuracy paves the way to a more reliable
comparison of the charge radii trends in the lead region.
