Tandem repeats (TRs) are sequences of DNA where two or more base pairs are repeated back-to-back at specific locations in the genome. The expansions of TRs are implicated in over 50 conditions, including Friedreich's ataxia, autism, and cancer. However, accurately measuring the copy number of TRs is challenging, especially when their expansions are larger than the fragment sizes used in standard short-read genome sequencing. Here we introduce ScatTR, a novel computational method that leverages a maximum likelihood framework to estimate the copy number of large TR expansions from short-read sequencing data. ScatTR calculates the likelihood of different alignments between sequencing reads and reference sequences that represent various TR lengths and employs a Monte Carlo technique to find the best match. In simulated data, ScatTR outperforms state-of-the-art methods, particularly for TRs with longer motifs and those with lengths that greatly exceed typical sequencing fragment sizes. When applied to data from the 1000 Genomes Project, ScatTR detected potential large TR expansions that other methods missed, highlighting its ability to better identify genome-wide characterization of TR variation. ScatTR can be accessed via: https://github.com/g2lab/scattr.