The local energy market (LEM) is a promising solution to address the imbalance problem caused by the increasing renewable energy. The network costs can serve as a price signal to guide nearby trading. However, the security-constrained economic dispatch method with large-scale prosumers has a lower calculation efficiency which fails to meet the high-frequency clearing needs of the LEM. Therefore, this paper proposes a fast sequential clearing method for congestion-aware local energy and flexibility markets. The LEM is designed based on continuous double auctions (CDA) to clear the market quickly. During CDA, network losses, and transmission fees are dynamically calculated based on AC power flow sensitivity factors and added to the bids/offers. To address congestion issues and prevent transaction cancellations, a local flexibility market (LFM) is established to utilize prosumers’ flexible capacity. The Market clearing efficiency of LFM is improved by adopting the fictitious nodal demand (FND) model and linear approximation of inscribed regular polygon constraint. The improved CDA in LEM promotes local transactions and enables effective network cost recovery. The LFM expands the total transaction volume by employing local flexibility. Case studies validate that the proposed methodology improves clearing efficiency, benefits prosumers, and ensures the operational security of distribution networks.