Preceding vehicles are easily lost due to the limited view of onboard sensors on roads with large curvature, resulting in the functional failure of the environmental perception system. This makes intelligent vehicles unable to obtain reliable information to make reasonable decisions, threatening the safety of car-following. Therefore, this paper proposes a method to improve car-following safety on blind curved roads. By analyzing the cautious mindset of drivers during cornering, an improved Markov model based on long-term prediction decay is proposed, combined with a Long Short-Term Memory (LSTM) model, to effectively predict the state of the preceding vehicle and address the intended functional safety issues arising from sensor functional failures. Subsequently, different strategies are designed for various states of the preceding vehicle, such as acceleration and deceleration, and a sensor field-of-view distance model is established to provide effective distance parameters for vehicle control. Notably, the proposed strategy has undergone rigorous simulation and hardware-in-the-loop (HiL) tests, proving its effectiveness in improving car-following safety on large curvature curves and instilling confidence in its potential.