The aggregation of alpha-synuclein into amyloid fibrils is a hallmark of Parkinson's disease. This process has been shown to directly involve interactions between proteins and lipid surfaces when the latter are present. Despite this importance, the molecular mechanisms of lipid-induced amyloid aggregation have remained largely elusive. Here, we present a global kinetic model to describe lipid-induced amyloid aggregation of alpha-synuclein. Using this framework we find that alpha-synuclein fibrils form via a two-step primary nucleation mechanism and that lipid molecules are directly involved in both the nucleation and fibril elongation steps, giving rise to lipid-protein coaggregates. To illustrate the applicability of this kinetic approach to drug discovery, we identify the mechanism of action of squalamine, a known inhibitor of alpha-synuclein aggregation, finding that this small molecule reduces the rate of lipid-dependent primary nucleation. Our work will likely guide the rational design of alpha-synuclein aggregation inhibitors.