Capacity estimation plays a crucial role in battery management systems, and is essential for ensuring the safety and reliability of lithium-sulfur (Li-S) batteries. This paper proposes a method that uses a long short-term memory (LSTM) neural network to estimate the state of health (SOH) of Li-S batteries. The method uses health features extracted from the charging curve and incremental capacity analysis (ICA) as input for the LSTM network. To enhance the robustness and accuracy of the network, the Adam algorithm is employed to optimize specific hyperparameters. Experimental data from three different groups of batteries with varying nominal capacities are used to validate the proposed method. The results demonstrate the effectiveness of the method in accurately estimating the capacity degradation of all three batteries. Also, the study examines the impact of different lengths of network training sets on capacity estimation. The results reveal that the ICA-LSTM model achieves a prediction accuracy of mean absolute error 4.6% and mean squared error 0.21% with three different training set lengths of 20%, 40%, and 60%. The analysis demonstrates that the lightweight model maintains high SOH estimation accuracy even with a small training set, and exhibits strong adaptive and generalization capabilities when applied to different Li-S batteries. Overall, the proposed method, supported by experimental validation and analysis, demonstrates its efficacy in ensuring accurate and reliable SOH estimation, thereby enhancing the safety and performance of Li-S batteries.