A streaming algorithm is adversarially robust if it is guaranteed to perform correctly even in the presence of an adaptive adversary. Recently, several sophisticated frameworks for robustification of classical streaming algorithms have been developed. One of the main open questions in this area is whether efficient adversarially robust algorithms exist for moment estimation problems under the turnstile streaming model, where both insertions and deletions are allowed. So far, the best known space complexity for streams of length $m$, achieved using differential privacy (DP) based techniques, is of order $\tilde{O}(m^{1/2})$ for computing a constant-factor approximation with high constant probability. In this work, we propose a new simple approach to tracking moments by alternating between two different regimes: a sparse regime, in which we can explicitly maintain the current frequency vector and use standard sparse recovery techniques, and a dense regime, in which we make use of existing DP-based robustification frameworks. The results obtained using our technique break the previous $m^{1/2}$ barrier for any fixed $p$. More specifically, our space complexity for $F_2$-estimation is $\tilde{O}(m^{2/5})$ and for $F_0$-estimation, i.e., counting the number of distinct elements, it is $\tilde O(m^{1/3})$. All existing robustness frameworks have their space complexity depend multiplicatively on a parameter $λ$ called the \emph{flip number} of the streaming problem, where $λ= m$ in turnstile moment estimation. The best known dependence in these frameworks (for constant factor approximation) is of order $\tilde{O}(λ^{1/2})$, and it is known to be tight for certain problems. Again, our approach breaks this barrier, achieving a dependence of order $\tilde{O}(λ^{1/2 - c(p)})$ for $F_p$-estimation, where $c(p) > 0$ depends only on $p$.