  We obtain exact Bardeen black holes to the regularized $4D$
Einstein--Gauss--Bonnet (EGB) gravity minimally coupled with the nonlinear
electrodynamics (NED). In turn, we analyze the horizon structure to determine
the effect of GB parameter $\alpha$ on the minimum cutoff values of mass,
$M_0$, and magnetic monopole charge, $g_0$, for the existence of a black hole
horizon. We obtain an exact expression for thermodynamic quantities, namely,
Hawking temperature $T_+$, entropy $S_+$, Helmholtz free energy $F_+$, and
specific heat $C_+$ associated with the black hole horizon, and they show
significant deviations from the $4D$ EGB case owing to NED. Interestingly,
there exists a critical value of horizon radius, $r_+^{c}$, corresponding to
the local maximum of Hawking temperature, at which heat capacity diverges,
confirming the second-order phase transition. A discussion on the black holes
of alternate regularized $4D$ EGB gravity belonging to the scalar-tensor theory
is appended.
