I feel like loading a new scene for every screen would add a lot of load times, and break the illusion of your character continuing his run from screen to screen.
If it were me, I would have the whole thing as one large level, with empty game objects in the position of where you want the camera for each screen. Have a trigger volume covering each individual screen, when the player enters the trigger it sets your cameras position to whatever given empty gameobject you positioned for that screen.