The bound-electron g factor is a stringent tool for tests of the standard model and the search for new physics. The comparison between an experiment on the g factor of lithiumlike silicon and the two recent theoretical values revealed the discrepancies of 1.7σ [Glazov et al. Phys. Rev. Lett. 123, 173001 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.173001] and 5.2σ [Yerokhin et al. Phys. Rev. A 102, 022815 (2020)PLRAAN2469-992610.1103/PhysRevA.102.022815]. To identify the reason for this disagreement, we accomplish large-scale high-precision computation of the interelectronic-interaction and many-electron QED corrections. The calculations are performed within the extended Furry picture of QED, and the dependence of the final values on the choice of the binding potential is carefully analyzed. As a result, we significantly improve the agreement between the theory and experiment for the g factor of lithiumlike silicon. We also report the most accurate theoretical prediction to date for lithiumlike calcium, which perfectly agrees with the experimental value.