Online attention is an increasingly valuable resource in the digital age, with extraordinary events such as the COVID-19 pandemic fuelling fierce competition around it. As misinformation pervades online platforms, users seek credible sources, while news outlets compete to attract and retain their attention. Here we measure the co-evolution of online ``engagement'' with (mis)information and its ``visibility'', where engagement corresponds to user interactions on social media, and visibility to fluctuations in user follower counts. Using over 100 million COVID-related retweets across 3 years, we analyse how user interactions and follower dynamics differ for factual, misleading and uncertain content. We observe that during major events (e.g., vaccine rollouts), users spreading factual content see rapid follower gain spikes, whereas those sharing misleading content tend to sustain faster growth outside of these high-attention periods. We introduce two scalable modelling frameworks (simple contagion and biased convergence) that reproduce many observed differing follower growth rates using temporal retweet network dynamics, providing evidence that content visibility co-evolves with user engagement. Our modelling lends itself to studying other large-scale events where online attention is at stake, such as climate and political debates.