  We propose a gauged $U(1)_{B-L}$ extended two Higgs doublet model to explain
both neutrino mass and lepton anomalous magnetic moments ($g-2$). Neutrino mass
is generated via an inverse seesaw mechanism by introducing singlet fermions.
Especially, we update the result of muon $g-2$ in light of the very recent
report by E989 experiment at Fermilab, indicating $a_\mu^{\rm
FNAL}=116592040(54)\times 10^{-11}$. Combining BNL result, we have the
following deviation from the standard model prediction $\Delta a_\mu=(2.51\pm
5.9)\times10^{-10}$ at 4.2 $\sigma$. Thanks to an appropriate assignment for
$U(1)_{B-L}\times Z_2$ symmetry and larger $(20\lesssim)\tan\beta$ that is
favoured by type-X model, we realize natural hierarchies among neutral
fermions. The lepton anomalous magnetic moments can be induced at the one loop
level by introducing an iso-spin singlet singly-charged boson. This charged
scalar plays a significant role in evading chiral suppression of these
phenomenologies. We show sizable lepton ($g-2$) can be obtained after
satisfying all the flavour constraints, such as $\mu\to e\gamma$ and flavour
conserving leptonic $Z$ boson decays.
