The global public sphere has changed dramatically over the past decades: a significant part of public discourse now takes place on algorithmically driven platforms owned by a handful of private companies. Despite its growing importance, there is scant large-scale academic research on the long-term evolution of user behaviour on these platforms, because the data are often proprietary to the platforms. Here, we evaluate the individual behaviour of 600,000 Twitter users between 2012 and 2019 and find empirical evidence for an acceleration of the way Twitter is used on an individual level. This manifests itself in the fact that cohorts of Twitter users behave differently depending on when they joined the platform. Behaviour within a cohort is relatively consistent over time and characterised by strong internal interactions, but over time behaviour from cohort to cohort shifts towards increased activity. Specifically, we measure this in terms of more tweets per user over time, denser interactions with others via retweets, and shorter content horizons, expressed as an individual's decaying autocorrelation of topics over time. Our observations are explained by a growing proportion of active users who not only tweet more actively but also elicit more retweets. These behaviours suggest a collective contribution to an increased flow of information through each cohort's news feed -- an increase that potentially depletes available collective attention over time. Our findings complement recent, empirical work on social acceleration, which has been largely agnostic about individual user activity.