The adapted Wasserstein ($AW$) distance refines the classical Wasserstein ($W$) distance by incorporating the temporal structure of stochastic processes. This makes the $AW$-distance well-suited as a robust distance for many dynamic stochastic optimization problems where the classical $W$-distance fails. However, estimating the $AW$-distance is a notably challenging task, compared to the classical $W$-distance. In the present work, we build a sharp estimate for the $AW$-distance in terms of the $W$-distance, for smooth measures. This reduces estimating the $AW$-distance to estimating the $W$-distance, where many well-established classical results can be leveraged. As an application, we prove a fast convergence rate of the kernel-based empirical estimator under the $AW$-distance, which approaches the Monte-Carlo rate ($n^{-1/2}$) in the regime of highly regular densities. These results are accomplished by deriving a sharp bi-Lipschitz estimate of the adapted total variation distance by the classical total variation distance.