Understanding dynamical complexity is one of the most important challenges in science. Significant progress has recently been made in optics through the study of dissipative soliton laser systems, where dynamics are governed by a complex balance between nonlinearity, dispersion, and energy exchange. A particularly complex regime of such systems is associated with noise-like pulse multiscale instabilities, where sub-picosecond pulses with random characteristics evolve chaotically underneath a much longer envelope. However, although observed for decades in experiments, the physics of this regime remains poorly understood, especially for highly-nonlinear cavities generating broadband spectra. Here, we address this question directly with a combined numerical and experimental study that reveals the physical origin of instability as nonlinear soliton dynamics and supercontinuum turbulence. Real-time characterisation reveals intracavity extreme events satisfying statistical rogue wave criteria, and both real-time and time-averaged measurements are in quantitative agreement with modelling.