Multi-modal Large Language Models (MLLMs) are increasingly deployed in interactive applications. However, their safety vulnerabilities become pronounced in multi-turn multi-modal scenarios, where harmful intent can be gradually reconstructed across turns, and security protocols fade into oblivion as the conversation progresses. Existing Reinforcement Learning from Human Feedback (RLHF) alignment methods are largely developed for single-turn visual question-answer (VQA) task and often require costly manual preference annotations, limiting their effectiveness and scalability in dialogues. To address this challenge, we present InterSafe-V, an open-source multi-modal dialogue dataset containing 11,270 dialogues and 500 specially designed refusal VQA samples. This dataset, constructed through interaction between several models, is designed to more accurately reflect real-world scenarios and includes specialized VQA pairs tailored for specific domains. Building on this dataset, we propose AM$^3$Safety, a framework that combines a cold-start refusal phase with Group Relative Policy Optimization (GRPO) fine-tuning using turn-aware dual-objective rewards across entire dialogues. Experiments on Qwen2.5-VL-7B-Instruct and LLaVA-NeXT-7B show more than 10\% decrease in Attack Success Rate (ASR) together with an increment of at least 8\% in harmless dimension and over 13\% in helpful dimension of MLLMs on multi-modal multi-turn safety benchmarks, while preserving their general abilities.