Sharding enhances blockchain scalability by partitioning nodes into multiple groups for concurrent transaction processing. Configuring a large number of small shards usually helps improve transaction concurrency, but it also increases the fraction of malicious nodes in each shard, easily causing shard corruption and jeopardizing system security. Existing works attempt to improve concurrency by reducing shard sizes while maintaining security, but typically rely on time-consuming recovery of corrupted shards to restore liveness and network-wide consensus. This causes severe system stagnation and limits scalability. To address this, we present StableShard, a sharded blockchain that securely provides high concurrency with stable and scalable performance. The core idea is to carefully co-design the division of labor between proposer shards (PSs) and finalizer committees (FCs): we deliberately assign 1) asymmetric roles and 2) matching parameters to PSs and FCs. Small PSs focus on fast transaction proposal and local validity, while large FCs focus on resolving forks, finalizing PS blocks, and maintaining liveness for faulty PSs via a cross-layer view-change protocol. Moreover, by fine-tuning key system parameters (e.g., shard size, quorum size), we ensure each PS to tolerate <1/2 fraction of malicious nodes without lossing liveness, and allow multiple FCs to securely coexist (each with <1/3 fraction of malicious nodes) for better scalability. Consequently, StableShard can safely configure many smaller PSs to boost concurrency, while FCs and PSs jointly guarantee safety and liveness without system stagnation, leading to stable and scalable performance. Evaluations show that StableShard achieves up to 10x higher throughput than existing solutions and significantly more stable concurrency under attacks.