In my campaigns I tend to break things into 'adventures,' each of which might take 2-4 sessions and then I have a good bit of downtime in between. If the party has been through enough, they gain a level during that downtime. I agree that experience is the key to learning, but practicing to perfection is important too. Adventures are the catalyst that gives them ideas for new techniques and spells, but they require some training and reflection to master them.

The added effect of the downtime is that it allows for the consequences of the party's actions to ripple out into the world and change things. There's just a lot more you can do with a story than spans months and years rather than a couple weeks. But if I were doing a pre-written module like OP, I'd probably just stick to that story and justify quick levels as best I could.

As a player, I try to show my character tinkering with or practicing features that they will gain next level. For example, a few times over a couple sessions, the party heard my Order of Scribes wizard talking to something with a deep voice but they never saw what it was. Naturally they were a bit unsettled. Then when I gained a level and got Manifest Mind, everyone was relieved to find I had just been talking to my now fully sentient spellbook.

Quote from Scarloc_Stormcall >>

When running a pre written campaign I generally never roleplay leveling, players just gain those new abilities and stats as per the level they are, most written campaigns keep the action rolling and so stopping for a long period of downtime to learn new skills and abilities is hard to work into the narrative. For example I have just finished running out of the abyss, for the first 2/3rds of that adventure the pc’s don’t stop constantly moving from one location to the next. 

In my homebrew campaigns it depends a little on the narrative and a lot on the players. Some players love to take a step back and RP downtime. If this is the case I will try and pace my adventure so major leveling, when a subclass gets a new ability generally, coincides with downtime, or a nice beat in the story. For instance the Arcane Trickster is close to level 9 I might coincide that level up with a session with another arcane trickster who has magical ambush, they offer to teach the rogue the ability through the session by the end the rogue knows the ability and has leveled up. This is easier with XP leveling as you can pace everyone’s leveling up over a series of sessions giving them all a chance to learn a new skill and spotlight a moment at the new level to show it off. 

With Wizards I have in the past told my wizard player as they approach the next major level to think about the next level spells they want to learn, I might then have situations arise where a spell scroll is found, or an NPC adds that new high level spell to there Izard spell book telling them they sense the character is almost ready to attempt it. This means we are not simply saying when the wizard levels up, you now learn 2 additional spells at your new level, what are they. It also gets the player thinking ahead of time about what they want to learn spell wise next. 

sorcerors, I will do a similar thing but I will instead roleplay it out. For instance Sorceror levels up and learns Fireball. The very first time they cast it I might say. 

“Looking at the creature advancing towards you you reach out for your trusty fire bolt spell, feeling the magic weave its way around your body coalescing at your hand the familiar feeling seems to grow, expanding to encase your very essence. Where you would normally feel the magic peak releasing the volt of fire from your hand it continues to grow. You stand, concentrating wondering just how much more energy you can cast into this spell. Finally you sense the power has reached its zenith, threatening to drain away you release it. A large ball of fire erupts from your hands, exploding on impact. You look at your hand, and then your colleagues. What felt like an eternity to you was a mere second, you stand, feeling drained, unsure where this new power came from you get a sense that, once rested, if required you could summon this new form of magic at will. 

 I look at it very similarly.  Leveling up is not a big deal.  I just tell them when we get to a town with training facilities you can train in the new skills and they pay money to the schools. It is more expensive for those who cast spells as they buy the new spells they learn, if they choose.