Accurate estimations of ice thickness and volume are indispensable for ice flow modelling, hydrological forecasts and sea-level rise projections. We present a new ice thickness estimation model based on a mass-conserving forward model and a Bayesian inversion scheme. The forward model calculates flux in an elevation-band flow-line model, and translates this into ice thickness and surface ice speed using a shallow ice formulation. Both ice thickness and speed are then extrapolated to the map plane. The model assimilates observations of ice thickness and speed using a Bayesian scheme implemented with a Markov chain Monte Carlo method, which calculates estimates of ice thickness and their error. We illustrate the model's capabilities by applying it to a mountain glacier, validate the model using 733 glaciers from four regions with ice thickness measurements, and demonstrate that the model can be used for large-scale studies by fitting it to over 30 000 glaciers from five regions. The results show that the model performs best when a few thickness observations are available; that the proposed scheme by which parameter-knowledge from a set of glaciers is transferred to others works but has room for improvements; and that the inferred regional ice volumes are consistent with recent estimates.