Multi-turn jailbreak attacks progressively erode LLM safety alignment across seemingly innocuous conversation turns, achieving success rates exceeding 90% against state-of-the-art models. Existing alignment-based and guardrail methods suffer from three key limitations: they require costly weight modification, evaluate each turn independently without modeling cumulative safety erosion, and detect attacks only after harmful content has been generated. To address these limitations, we first formulate the proactive early jailbreak detection problem with a new metric, detection lead, that measures how early an attack can be detected before the LLM complies. We then propose SAFEDREAM, a lightweight world-model-based framework that operates as an external module without modifying the LLM's weights. SAFEDREAM introduces three components: (1) a safety state world model that encodes LLM hidden states into a compact safety representation and predicts how it evolves across turns, (2) CUSUM detection that accumulates weak per-turn risk signals into reliable evidence, and (3) contrastive imagination that simultaneously rolls out attack and benign futures in latent space to issue early alarms before jailbreaks occur. On three multi-turn jailbreak benchmarks (XGuard-Train, SafeDialBench, SafeMTData) against 8 baselines, SAFEDREAM achieves the best detection timeliness across all benchmarks (1.06-1.20 turns before compliance) while maintaining competitive false positive rates and outperforming baselines in detection quality.