The "shopping mall" level used to be done with fire of some sort, but it's not only slower by a few seconds, but also harder.
Ideally, you still have the lightning rod from the previous level. As you pass through the first loading zone, put the powerup in your mouth, and in the next room, spit it out on the two Poppy Bros. If you don't have a powerup, it's fastest to just spit one into the other. On the next conveyer belt, headbutt the second Bouncy. To get around the Putt's boulder on the one after, there are two main strats: one where you puff the Putt and jump up the platforms, and one where you float past it. I believe the former is slightly faster. After grabbing Shard 1, jump a little before the ledge so you land straight on the conveyer belt. Jump past the other Putt with a little float and puff and take the elevator.
On the first floor of the next room, there are three Mahalls and two Sawyers. Either you can form a double-star and kill all the Mahalls at once, or dodge all of the enemies carefully. I think the double-star strat is faster because you're able to slide three times. A really simple way of forming the star, as shown in the middle, is to stand underneath the first Mahall and start inhaling, but this animation takes a long time.
The second floor consists of three Gloms and two Fishbones. Here, three strats are shown for getting around these enemies. I find the first strat to be most consistent: grinding against the wall while jumping allows you to just hold left the entire time. However, the third strat allows you to get some food if you're low on health.
In the first strat, get one of the Ticks and use the needle powerup to get past the I³ enemies. For a tiny framesaver, you can try puffing a Tick instead of sliding into it, but the timing is really tight. The second I³ can be slowly jumped over (with the help of a puff), so you don't have to take extra damage there. A simpler strat is to inhale the second Tick and swallow it while waiting for the first I³ is falling. The rest of the floor proceeds the same way as the previous strat.
The miniboss is a straightforward fight where only three attacks are needed—each use of needle can hit twice. Grab Shard 2 after the fight. The Bo enemy on the right can move randomly in your way, so you might jump into the shard while using needle. In the next room, press the switches corresponding to the colors of the fruit Adeleine draws. The fruits are, in the order of the switches in the next room, cherry, peach, nut(?), melon, grape, lemon, orange. I like to do a needle into the wall to get in position to slide, and I start the slide as Adeleine draws the third fruit. Shard 3 spawns in the middle above the green switch, and depending on which switches you pressed, you might be able to slide off of one of them to get to the door quicker.
In the first section, you can just run underneath everyone, but you can save a bit of time by sliding off the platforms. The first slide is not really worth it since you have to puff into the Kiyobi's spear, but I think the second slide is definitely worth going for. Afterwards, in the room with the falling bookcases, take out needle and run into the Cairn to get drill. As with 3-3, jump out of the water repeatedly in the fishtank. I like to use drill as I jump out of the water to get a small boost and take care of the first Sawyer in the next room. Use drill as shown in the bottom-right to take out some inconvenient enemies.
The Shotzos in the last room will try to predict where you go. As long as you route out your jumps, you shouldn't ever get hit.