In the previous level, you should end with stone. Backtrack all the way to 2-4 to get dynamite. If you don't have stone (e.g. if you ran into the enemy at the end of 4-1 with fire), backtrack to 1-2 to get both abilities. Keep in mind that this powerup needs to be kept almost the whole level, so be careful about not losing it.
Shard 1 is in the first room, just right of the waterfall on the left. For some reason, everyone starts throwing dynamites after grabbing it, including me. The next room is a minecart autoscroller, where jumping repeatedly saves a bit of time. Unlike the river autoscroller, sometimes the track splits into two. By always taking the top path, you'll soon find Shard 2 sitting on top of an obstacle. To make sure I don't crash into the rock, my strat is to do a small jump at the kink in the track.
As you're falling, throw a dynamite to avoid faceplanting. Then, run forward to headbutt the dynamite backwards to avoid having it explode on you. The timing for this is really really generous. Afterwards, jump over all the I³ enemies. If timed correctly, there is no need for any puffing. After dodging the two Gloms, puff one of the blocks to get through. It's possible to squeeze in the one-tile gap, but I don't think it's worth trying for. Since the block is on the ground, another possibility is to slide into it, but this is slower. The I³ is surrounded by Gordos, so the fastest way to get past it is to slide underneath and take damage, boosting you to the other side. An alternative that doesn't involve damaging yourself is to wait and then jump over it (shown on the right).
The next room has flowing water, so as usual, jump repeatedly on water moving to the left, and run down water moving to the right, even if it's sloped. Be careful not to jump into any of the enemies on the ceiling. The last section of this room (right) can be done by only running on the water going to the right.
Most of this section involves jumping at the right time to manipulate the Kapar's jump height. At the first cliff, do a medium-sized jump to dodge the Zoos's attack and fall in between the blocks without faceplanting. The next Zoos enemy will often be in a position to hit you, so slow down just slightly (e.g. by turning around twice) to avoid its attack.
Shard 3 is at the bottom of the huge cliff, and throwing a dynamite into the nearest enemy is just close enough to break the barrier (middle). To protect yourself from the blast, use the safety hat by pressing down after the dynamite has been thrown. Note that missing the Squibby enemy will waste a lot of time since you have to wait for the dynamite to explode. If you hit the rightmost point of its hitbox, the dynamite will actually be too far away from the barrier to break it. To combat these difficulties, I came up with a little setup that involves running off the cliff before throwing it (right).
The miniboss fight has six sets of four Drop enemies. For each set, you have to kill them one by one, so optimally it will look like four waves of six enemies. Luckily, the dynamite powerup is great for this fight. I try to hit an enemy closer to the middle to ensure that it hits all six at once. Depending on the health situation, I might jump up and anticipate the spawn for a slightly quicker kill. Since you can't use the hard hat in the air, the dynamite can damage boost you towards the door on the right. The next room has a Maxim Tomato at the beginning so ending the fight with low health is not a problem.
Most of this final room is straightforward, since a lot of the time you'll be waiting for these blue platform cycles. While waiting, you can discard your powerup. For one set of platforms, the timing is a bit tight, but sliding off ledges makes it a lot easier. There is one Gabon in the way while waiting for a platform (right), so swallow his skull hat. If done quickly enough, he'll run into the platforms himself. Otherwise, he'll rush at you and you'll have to spit his hat back out at him. The last blue platform isn't on a timer. Instead you have to approach it to get it to move.