This work considers the fundamental privacy limits under active fingerprinting attacks in power-law bipartite networks. The scenario arises naturally in social network analysis, tracking user mobility in wireless networks, and forensics applications, among others. A stochastic growing network generation model -- called the popularity-based model -- is investigated, where the bipartite network is generated iteratively, and in each iteration vertices attract new edges based on their assigned popularity values. It is shown that using the appropriate choice of initial popularity values, the node degree distribution follows a power-law distribution with arbitrary parameter $α>2$, i.e. fraction of nodes with degree $d$ is proportional to $d^{-α}$. An active fingerprinting deanonymization attack strategy called the augmented information threshold attack strategy (A-ITS) is proposed which uses the attacker's knowledge of the node degree distribution along with the concept of information values for deanonymization. Sufficient conditions for the success of the A-ITS, based on network parameters, are derived. It is shown through simulations that the proposed attack significantly outperforms the state-of-the-art attack strategies.