Large language models (LLMs) excel at single-turn reasoning but often lose accuracy and coherence over extended, multi-turn interactions. Recent evaluations such as TurnBench highlight recurring failure modes, including reasoning bias, task drift, hallucination, overconfidence, and memory decay. Current approaches typically append full conversational histories, causing unbounded context growth, higher computational costs, and degraded reasoning efficiency. We introduce CogMem, a cognitively inspired, memory-augmented LLM architecture that supports sustained iterative reasoning through structured, persistent memory. CogMem incorporates three layers: a Long-Term Memory (LTM) that consolidates cross-session reasoning strategies; a Direct Access (DA) memory that maintains session-level notes and retrieves relevant long-term memories; and a Focus of Attention (FoA) mechanism that dynamically reconstructs concise, task-relevant context at each turn. Experiments on TurnBench show that this layered design mitigates reasoning failures, controls context growth, and improves consistency across extended reasoning chains, suggesting that structured hierarchical memory is a promising direction for improving long-horizon multi-turn reasoning in large language models.