Blockchain databases have attracted widespread attention but suffer from poor scalability due to underlying non-scalable blockchains. While blockchain sharding is necessary for a scalable blockchain database, it poses a new challenge named on-chain cross-shard database services. Each cross-shard database service (e.g., cross-shard queries or inter-shard load balancing) involves massive cross-shard data exchanges, while the existing cross-shard mechanisms need to process each cross-shard data exchange via the consensus of all nodes in the related shards (i.e., on-chain) to resist a Byzantine environment of blockchain, which eliminates sharding benefits. To tackle the challenge, this paper presents GriDB, the first scalable blockchain database, by designing a novel off-chain cross-shard mechanism for efficient cross-shard database services. Borrowing the idea of off-chain payments, GriDB delegates massive cross-shard data exchange to a few nodes, each of which is randomly picked from a different shard. Considering the Byzantine environment, the untrusted delegates cooperate to generate succinct proof for cross-shard data exchanges, while the consensus is only responsible for the low-cost proof verification. However, different from payments, the database services' verification has more requirements (e.g., completeness, correctness, freshness, and availability); thus, we introduce several new authenticated data structures (ADS). Particularly, we utilize consensus to extend the threat model and reduce the complexity of traditional accumulator-based ADS for verifiable cross-shard queries with a rich set of relational operators. Moreover, we study the necessity of inter-shard load balancing for a scalable blockchain database and design an off-chain and live approach for both efficiency and availability during balancing.