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.