Job interviews play a critical role in shaping one's career, yet practicing interview skills can be challenging, especially without access to human coaches or peers for feedback. Recent advancements in large language models (LLMs) present an opportunity to enhance the interview practice experience. Yet, little research has explored the effectiveness and user perceptions of such systems or the benefits and challenges of using LLMs for interview practice. Furthermore, while prior work and recent commercial tools have demonstrated the potential of AI to assist with interview practice, they often deliver one-way feedback, where users only receive information about their performance. By contrast, dialogic feedback, a concept developed in learning sciences, is a two-way interaction feedback process that allows users to further engage with and learn from the provided feedback through interactive dialogue. This paper introduces Conversate, a web-based application that supports reflective learning in job interview practice by leveraging large language models (LLMs) for interactive interview simulations and dialogic feedback. To start the interview session, the user provides the title of a job position (e.g., entry-level software engineer) in the system. Then, our system will initialize the LLM agent to start the interview simulation by asking the user an opening interview question and following up with questions carefully adapted to subsequent user responses. After the interview session, our back-end LLM framework will then analyze the user's responses and highlight areas for improvement. Users can then annotate the transcript by selecting specific sections and writing self-reflections. Finally, the user can interact with the system for dialogic feedback, conversing with the LLM agent to learn from and iteratively refine their answers based on the agent's guidance.