The aerodynamic design of turbomachinery is a complex and tightly coupled multi-stage process involving geometry generation, performance prediction, optimization, and high-fidelity physical validation. Existing intelligent design approaches typically focus on individual stages or rely on loosely coupled pipelines, making fully autonomous end-to-end design challenging. To address this issue, this study proposes TurboAgent, a large language model (LLM)-driven autonomous multi-agent framework for turbomachinery aerodynamic design and optimization. The LLM serves as the core for task planning and coordination, while specialized agents handle generative design, rapid performance prediction, multi-objective optimization, and physics-based validation. The framework transforms traditional trial-and-error design into a data-driven collaborative workflow, with high-fidelity simulations retained for final verification. A transonic single-rotor compressor is used for validation. The results show strong agreement between target performance, generated designs, and CFD simulations. The coefficients of determination for mass flow rate, total pressure ratio, and isentropic efficiency all exceed 0.91, with normalized RMSE values below 8%. The optimization agent further improves isentropic efficiency by 1.61% and total pressure ratio by 3.02%. The complete workflow can be executed within approximately 30 minutes under parallel computing. These results demonstrate that TurboAgent enables an autonomous closed-loop design process from natural language requirements to final design generation, providing an efficient and scalable paradigm for turbomachinery aerodynamic design.