The newness of the players is also a factor. While I love going to Level 2 after 1 session, I might go a little bit longer if there's someone new to the game (there's enough decision points in a single session that to immediately give them more can feel like a lot!). With a more experienced group, I might start at 3rd level, so the timeline between 3 and 4 is much shorter than between 1 and 4!
But generally:
1 session to get to 2nd
2 sessions to get to 3rd
3 sessions to get to 4th.
~ 2 months of real-world time. I don't use monster XP, so I typically set my milestones to significant story events, and pace the significant story events similarly (one in the first session, one in the next 2 sessions, then one in the next 3 sessions). A Tier 1 story arc usually has 4 significant beats to get to LV 5.
A simple example might be something like...
Session 1: Beginning: There's some bandits working with some goblins.
Session 2-3: Rising Action: Goblins and Hobgoblins and Bugbears are all supporting the local banditry.
Session 3-6: Moment of Crisis: A Hobgoblin King is coming to take the throne for himself. Their army is spotted, and it's thousands strong.
Session 6-9: Climax: The Hobgoblin King and the armies of the PC's home kingdom clash. The party is sent to eliminate the King and his two most powerful generals (one goblin, one bugbear). If these three are taken out, the hobgoblin army will route as the goblins and bugbears rebel and the hobgoblins fall to infighting to claim the empty throne.