Over the past two decades, the number of close social connections increased substantially, at least by a factor of two. At the same time, societal opinions have become increasingly polarized in many Western countries. To explore whether these trends could be connected, we employ a simple computational model of society, where people—within their social networks—continuously compare and update their opinions. Here, we show that the model that is known to realistically capture both homophily and social balance exhibits a phase transition phenomenon where, above a critical social connectivity, an explosive transition toward strong polarization must occur. The model allows us to understand the empirical inflation of polarization during the last decades as a function of the observed increased values of social connectivity. In the presence of a small fraction of synchronized influencers, the transition becomes continuous; however, polarization then appears at lower connectivities. We discuss the implications of the presence of a phase transition in social polarization.