Recent text-to-SQL models have achieved strong performance, but their effectiveness remains largely confined to SQLite due to dataset limitations. However, real-world applications require SQL generation across multiple dialects with varying syntax and specialized features, which remains a challenge for current models. The main obstacle in building a dialect-aware model lies in acquiring high-quality dialect-specific data. Data generated purely through static prompting—without validating SQLs via execution—tends to be noisy and unreliable. Moreover, the lack of real execution environments in the training loop prevents models from grounding their predictions in executable semantics, limiting generalization despite surface-level improvements from data filtering. This work introduces ExeSQL, a text-to-SQL framework with execution-driven, agentic bootstrapping. The method consists of iterative query generation, execution-based filtering (e.g., rejection sampling), and preference-based training, enabling the model to adapt to new SQL dialects through verifiable, feedback-guided learning. Experiments show that ExeSQL bridges the dialect gap in text-to-SQL, achieving average improvements of 15.2%, 10.38%, and 4.49% over GPT-4o on PostgreSQL, MySQL, and Oracle, respectively, across multiple datasets of varying difficulty.