The distribution of fitness effects (DFE) of new beneficial mutations is a key quantity that dictates the dynamics of adaptation. The recently developed barcode lineage tracking (BLT) approach is an important advance toward measuring DFEs. BLT experiments enable researchers to track the frequencies of ~105 of barcoded lineages in large microbial populations and detect up to thousands of nascent beneficial mutations in a single experiment. However, reliably identifying adapted lineages and estimating the fitness effects of the driver mutations remains a challenge because lineage dynamics are subject to demographic and measurement noise and competition with other lineages. We show that the currently standard Levy-Blundell method for analyzing BLT data can be biased in certain regimes. To address this problem, we develop a new method called BASIL (BAyesian Selection Inference for Lineage tracking data), which dynamically updates the belief distribution of each lineage\'s fitness and size based on the number of barcode reads. We calibrate BASIL\'s model of noise with new experimental data and find that noise variance scales non-linearly with lineage abundance. We test BASIL\'s performance on simulated data and data from published BLT studies and find that it is both robust and accurate. Finally, we characterize how BASIL\'s power to identify adapted lineages depends on the frequency of sampling and read coverage. Our work paves the way for a systematic inference of the distribution of fitness effects of new beneficial mutations from BLT experiments in a variety of scenarios.