Traffic count (or link count) data represents the cumulative traffic in the lanes between two consecutive signalised intersections. Typically, dedicated infrastructure-based sensors are required for link count data collection. The lack of adequate data collection infrastructure leads to lack of link count data for numerous cities, particularly those in low- and middle-income countries. Here, we address the research problem of link count estimation using crowd-sourced trajectory data to reduce the reliance on any dedicated infrastructure. A stochastic queue discharge model is developed to estimate link counts at signalised intersections taking into account the sparsity and low penetration rate (i.e., the percentage of vehicles with known trajectory) brought on by crowdsourcing. The issue of poor penetration rate is tackled by constructing synthetic trajectories entirely from known trajectories. The proposed model further provides a methodology for estimating the delay resulting from the start-up loss time of the vehicles in the queue under unknown traffic conditions. The proposed model is implemented and validated with real-world data at a signalised intersection in Kolkata, India. Validation results demonstrate that the model can estimate link count with an average accuracy score of 82% with a very low penetration rate (not in the city, but at the intersection) of 5.09% in unknown traffic conditions, which is yet to be accomplished in the current state-of-the-art.