Crowding effects are key to the self-organization of densely packed cellular assemblies, such as biofilms, solid tumors, and developing tissues. When cells grow and divide they push each other apart, remodeling the structure and extent of the populations range. It has recently been shown that crowding effects also couple the evolutionary fate of neighboring cells, thereby weakening the strength of natural selection. However, the impact of crowding on neutral processes remains unclear. Here, we quantify the genetic diversity of expanding microbial colonies and uncover signatures of crowding in the site frequency spectrum. By combining fluctuation tests, cell-based simulations, and lineage tracing in a novel microfluidic incubator, we find that the majority of mutations arise behind the expanding frontier, giving rise to clones that are mechanically "pushed out" of the growing region by the proliferating cells in front. These excluded-volume interactions result in a clone size distribution that solely depends on where the mutation first arose relative to the front and is characterized by a simple power-law for sizes below a critical threshold. Our model and simulations predict that the distribution only depends on a single parameter, the characteristic growth layer thickness, and hence allows estimation of the mutation rate in a variety of crowded cellular populations. Combined with previous studies on high-frequency mutations, our finding provides a unified picture of the genetic diversity in expanding populations over the whole frequency range and suggests a practical method to assess growth dynamics by sequencing populations across scales.