The first 3 levels go very quickly. They're supposed to. You basically gain a level a session for the first 3 levels, and then it starts to slow down.
Check the xp chart.
This.
The game is set up such that level 3 is your "I'm now an adventurer" level, as that's when most classes get their subclass. So levels 1 and 2 are the "introduction to adventuring" levels and just by looking at the XP chart, are meant to get through in pretty much a single session or two for each. So even if you were doing XP calculation rather than milestone... reaching level 3 by like the 3rd session is expected and the way they set the game up.
After that... Level 5 is the next leveling milestone and when adventurers "get their black belt" as it were. Spellcasters get their 3rd level spells, martial characters get their second attack, and everyone's proficiency bonus goes up to +3... which is what the game uses to signify them as Experienced Adventurers. Thus, the XP to get through levels 3 and 4 to reach 5 are also not too bad. However, it's the path from 5 to 11 that sees the slowdown in level gaining. This is where multiple sessions over the normal advancement rate will occur, as the game intentionally slows things down because it's where traditionally most groups hang out in.
But then once they reach 11th level and get out of the adventurer's tier, they move into the paragon tier and again leveling speeds up a bit. This was done because so many groups wrap their games up prior to 11th that they wanted to keep those games that reach this high moving forward with propulsion. So they made advancement again a bit easier just to keep groups moving forward.
Now obviously if you are doing an Adventurer's Path, the book is set up for you to level when appropriate for the story. So if the group streams through the chapters of the story quickly, then leveling will go up quickly too. At that point you either (as was mentioned above) throw in extra encounters so that the party has more things to do before reaching the next chapter, or you don't level them up at the AP's designed chapter breaks (but then run the risk of them being under-leveled and under-powered for future chapters.) But if you happen to have a larger than normal size of a group, or if you are a bit more generous with magic items and the like... the risk of them being under-powered for later chapters if you slowed down your milestone leveling is lessened.
Personally... I'd just have fun with the book by adding in additional encounters, especially roleplaying ones. I've had sessions where so much roleplaying has occurred that they naturally slowed down their own milestone advancement. You end up having an extra session or two in the milestone leveling because the party spent sessions talking with people, rather than hitting all the requisite encounters needed to advance to the next chapter of the story. Throw in an extra fight or two and you'll find things even slower.