  A rigorous QED evaluation of the two-photon exchange corrections to the g
factor of lithiumlike ions is presented. The screened self-energy corrections
are calculated for the intermediate-$Z$ region and its accuracy for the
high-$Z$ region is essentially improved in comparison with that of previous
calculations. As a result, the theoretical accuracy of the g factor of
lithiumlike ions is significantly increased. The theoretical prediction
obtained for the g factor of $^{28}$Si$^{11+}$ ${\rm g}_{\rm th} =
2.000\,889\,892(8)$ is in an excellent agreement with the corresponding
experimental value ${\rm g}_{\rm exp} = 2.000\,889\,889\,9(21)$ [A. Wagner et
al., Phys. Rev. Lett. 110, 033003 (2013)].
