Globalization has led to the introduction of thousands of alien species worldwide. With growing impacts by invasive species, understanding the invasion process remains critical for predicting adverse effects and informing efficient management. Theoretically, invasion dynamics have been assumed to follow an “invasion curve” (S‐shaped curve of available area invaded over time), but this dynamic has lacked empirical testing using large‐scale data and neglects to consider invader abundances. We propose an “impact curve” describing the impacts generated by invasive species over time based on cumulative abundances. To test this curve's large‐scale applicability, we used the data‐rich New Zealand mud snail Potamopyrgus antipodarum, one of the most damaging freshwater invaders that has invaded almost all of Europe. Using long‐term (1979–2020) abundance and environmental data collected across 306 European sites, we observed that P. antipodarum abundance generally increased through time, with slower population growth at higher latitudes and with lower runoff depth. Fifty‐nine percent of these populations followed the impact curve, characterized by first occurrence, exponential growth, then long‐term saturation. This behaviour is consistent with boom‐bust dynamics, as saturation occurs due to a rapid decline in abundance over time. Across sites, we estimated that impact peaked approximately two decades after first detection, but the rate of progression along the invasion process was influenced by local abiotic conditions. The S‐shaped impact curve may be common among many invasive species that undergo complex invasion dynamics. This provides a potentially unifying approach to advance understanding of large‐scale invasion dynamics and could inform timely management actions to mitigate impacts on ecosystems and economies. Invasion dynamics have been assumed to follow an “invasion curve”, which is a S‐shaped curve of available area invaded over time, but which has lacked empirical testing. Testing this curve’s large‐scale applicability using a large spatiotemporal dataset of the invasive New Zeland mud snail, we identified 59% of its populations to follow this curve. These population dynamics can be used to represent its impact on invaded ecosystems over time. Thus, the S‐shaped impact curve may be common among many invasive species that undergo complex invasion dynamics.