We show that the doping-controlled superconductor-insulator transition (SIT) in a high critical temperature cuprate system (${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$) exhibits a fundamentally different behavior than is expected from conventional SIT. At the critical doping, the sheet resistance seems to diverge in the zero-temperature limit. Above the critical doping, the transport is universally scaled by a two-component conductance model. Below, it continuously evolves from weakly to strongly insulating behavior. The two-component conductance model suggests that a collective electronic phase-separation mechanism may be responsible for this unconventional SIT behavior.