Over the past decade, the AI industry has come to exert an unprecedented economic, political and societal power and influence. It is therefore critical that we comprehend the extent and depth of pervasive and multifaceted capture of AI regulation by corporate actors in order to contend and challenge it. In this paper, we first develop a taxonomy of mechanisms enabling capture to provide a comprehensive understanding of the problem. Grounded in design science research (DSR) methodologies and extensive scoping review of existing literature and media reports, our taxonomy of capture consists of 27 mechanisms across five categories. We then develop an annotation template incorporating our taxonomy, and manually annotate and analyse 100 news articles. The purpose behind this analysis is twofold: validate our taxonomy and provide a novel quantification of capture mechanisms and dominant narratives. Our analysis identifies 249 instances of capture mechanisms, often co-occurring with narratives that rationalise such capture. We find that the most recurring categories of mechanisms are Discourse & Epistemic Influence, concerning narrative framing, and Elusion of law, related to violations and contentious interpretations of antitrust, privacy, copyright and labour laws. We further find that Regulation stifles innovation, Red tape and National Interest are the most frequently invoked narratives used to rationalise capture. We emphasize the extent and breadth of regulatory capture by coalescing forces -- Big AI and governments -- as something policy makers and the public ought to treat as an emergency. Finally, we put forward key lessons learned from other industries along with transferable tactics for uncovering, resisting and challenging Big AI capture as well as in envisioning counter narratives.