We investigate the interaction between the counter-rotating bicircular field and the trivial and topological insulator with anomalous Hall conductivity (AHC) to show the effect of the asymmetric spin band and topological invariant. We show that the reaction of the system to the counter-rotating bicircular field is classified into the high-field and low-field regimes. In the high-field regime, it is shown that the AHC of the system is controlled by the phase difference between the ω0 and 2ω0 fields. We also show that in the low-field regime, the AHC of the topological insulator is determined by the helicity of the laser, while the AHC is negligible in the trivial insulator. For the spin-orbit coupling (SOC), it is demonstrated that the high SOC increases the required field amplitude for the transition from the low-field to the high-field regime. Also, we show that strong SOC leads to an additional sign change of the AHC in the high-field regime, but with different origins in the trivial and topological insulator.