That's the expectation. The adventures paths assume characters will use the Medium XP track for progression, and assuming they hit all the major points and a fair amount of the secondary encounters, they should consistently remain in the appropriate level range for the next adventure.
But it's up to the GM to keep an eye on that and make sure that if they're falling behind, something is done. In at least some adventure paths, this can be done easily by inserting a side adventure between the AP modules. In others it may be a bit more awkward to justify the presence of such a side adventure, but it's definitely doable (or you can just up the number of random encounters for a while, or provide larger story awards, in the worst case).
In general, yes, the AP ensures enough XP to hit target levels assuming the players keep more or less to the path. APs will often state to give XP for circumventing events (for example, the players manage to figure out the murder mystery and arrest the bad guy before he hires an assassin, you still give the players the XP they would have received from killing the assassin). It is still possible for players to choose to bypass sections of some APs and miss out on XP. That's when you as the GM should be watching and either give them bonus xp (either a "congrats on finishing book 2" or a small boost to the rest of the book's xp).
Now, that said, GMs have started switching to a "you level up when I say so" methodology for APs. So in Carrion Crown book 1, the players come to a town, learn about the haunted prison, deal with bad things happening in town, then clean out the ghosts. So you say players start at level 1, gain level 2 when they have learned about the prison and talked with the locals, hit 3 when they have dealt with all the bad things in town, and gain level 4 when they clear out the prison and finish the AP.
You have to be a bit fluid about the gates, but the AP usually says the players should be at X when they start, Y when they get to this place, and Z when they are done. Use those markers and it'll go easily.
The benefit is that players aren't hunting for XP and trying to grind levels by wandering around the woods hoping for random encounters. The downside is that you can't use XP as an ad hoc reward and players can't see how close they are to leveling.
The adventures offer guidance on where players ought to level up, but like any adventure that is not a pure railroad, there is no gaurantee that the players will want to or need to overcome every adversary the ap lays out. Therefore its neccessary for the GM to keep an eye on these things and either add in encounters, provide extra xp, or simply drop xp and level the players where the AP says (alot of dms go for that nowadays).