Combining the multi-lepton searches at the LHC, we study the possibilities of accommodating the new data of muon and electron $g-2$ anomalies in the lepton-specific inert two-Higgs-doublet model. We take the heavy CP-even Higgs as the 125 GeV Higgs, and find the muon and electron $g-2$ anomalies can be explained simultaneously in the region of 5 GeV $< m_h<60$ GeV, 200 GeV $<m_A< 620$ GeV, 190 GeV $<m_{H^pm}< 620$ GeV for appropriate Yukawa couplings between leptons and inert Higgs. Meanwhile, the model can give a better fit to the data of lepton universality in $tau$ decays than the SM. Further, the multi-lepton event searches at the LHC impose a stringent upper bound on $m_h$, $m_h<$ 35 GeV.
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