Control of interfacial morphology in electrochemical processes is essential for applications ranging from nanomanufacturing to batteries. Here, we quantify the evolution of an electrochemical growth front, using liquid cell electron microscopy to access unexplored length and time scales. During galvanostatic deposition of copper from an acidic electrolyte, we find that the growth front initially evolves consistent with kinetic roughening theory. Subsequently, it roughens more rapidly, consistent with diffusion-limited growth physics. However, the onset of roughening is strongly delayed compared to expectations, suggesting the importance of lateral diffusion of ions. Based on these growth regimes, we discuss morphological control and demonstrate the effects of two strategies, pulse plating and the use of electrolyte additives.