Medical question-answering benchmarks predominantly evaluate single-turn exchanges, failing to capture the iterative, clarification-seeking nature of real patient consultations. We introduce ThReadMed-QA, a benchmark of 2,437 fully-answered patient-physician conversation threads extracted from r/AskDocs, comprising 8,204 question-answer pairs across up to 9 turns. Unlike prior work relying on simulated dialogues, adversarial prompts, or exam-style questions, ThReadMed-QA captures authentic patient follow-up questions and verified physician responses, reflecting how patients naturally seek medical information online. We evaluate five state-of-the-art LLMs -- GPT-5, GPT-4o, Claude Haiku, Gemini 2.5 Flash, and Llama 3.3 70B -- on a stratified test split of 238 conversations (948 QA pairs) using a calibrated LLM-as-a-judge rubric grounded in physician ground truth. Even the strongest model, GPT-5, achieves only 41.2% fully-correct responses. All five models degrade significantly from turn 0 to turn 2 (p < 0.001), with wrong-answer rates roughly tripling by the third turn. We identify a fundamental tension between single-turn capability and multi-turn reliability: models with the strongest initial performance (GPT-5: 75.2; Claude Haiku: 72.3 out of 100) exhibit the steepest declines by turn 2 (dropping 16.2 and 25.0 points respectively), while weaker models plateau or marginally improve. We introduce two metrics to quantify multi-turn failure modes: Conversational Consistency Score (CCS) and Error Propagation Rate (EPR). CCS reveals that nearly one in three Claude Haiku conversations swings between a fully correct and a completely wrong response within the same thread. EPR shows that a single wrong turn raises the probability of a subsequent wrong turn by 1.9-6.1x across all models.