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In the second-order response regime, the Hall voltage can be nonzero without breaking the time-reversal symmetry, as long as the system is noncentrosymmetric. There are multiple mechanisms with different scaling rules that contribute to the nonlinear Hall effect (NLHE). The intrinsic contribution is closely related to the Berry curvature dipole and has been extensively investigated recently. The study of the extrinsic contribution, however, is scarce, although it can enter the NLHE even in the leading order. Here, we report a giant nonlinear transport response in TBG, in which the intrinsic mechanism is forbidden. The magnitude and direction of the second-order nonlinearity can be effectively tuned by the gate voltage. The peak value of the second-order Hall conductivity close to the full filling of the moiré band reaches 8.76 $μmSV^{-1}$, four-order larger than those detected in $WTe_2$. The observed giant second-order nonlinearity can be understood from the collaboration of the asymmetric scattering of electrons off the static (Coulomb impurities) and dynamic disorders (phonons) in noncentrosymmetric crystals. It is mainly determined by the skew-scattering contribution from impurities at 1.7 K. The skew-scattering from phonons has a much larger coupling coefficient as suggested by the scaling results, and becomes as important as the impurity contribution as the temperature rises. Our observations demonstrate the potential of TBG in studying nonlinear response and possible rectification applications.