The modern in-memory database (IMDB) can support highly concurrent on-line transaction processing (OLTP) workloads and generate massive transactional logs per second. Quorum-based replication protocols such as Paxos or Raft have been widely used in the distributed databases to offer higher availability and fault-tolerance. However, it is non-trivial to replicate IMDB because high transaction rate has brought new challenges. First, the leader node in quorum replication should have adaptivity by considering various transaction arrival rates and the processing capability of follower nodes. Second, followers are required to replay logs to catch up the state of the leader in the highly concurrent setting to reduce visibility gap. Third, modern databases are often built with a cluster of commodity machines connected by low configuration networks, in which the network anomalies often happen. In this case, the performance would be significantly affected because the follower node falls into the long-duration exception handling process (e.g., fetch lost logs from the leader). To this end, we build QuorumX, an efficient and stable quorum-based replication framework for IMDB under heavy OLTP workloads. QuorumX combines critical path based batching and pipeline batching to provide an adaptive log propagation scheme to obtain a stable and high performance at various settings. Further, we propose a safe and coordination-free log replay scheme to minimize the visibility gap between the leader and follower IMDBs. We further carefully design the process for the follower node in order to alleviate the influence of the unreliable network on the replication performance. Our evaluation results with the YCSB, TPC-C and a realistic micro-benchmark demonstrate that QuorumX achieves the performance close to asynchronous primary-backup replication and could always provide a stable service with data consistency and a low-level visibility gap.