If you go through the exit the game normally expects you to go through after getting the Leaf, you can [Item Slide] into the entrance of Forbidden Woods by backing up against the left wall, charging an [Item Slide] against it and then slash forwards. Unlike what the video says, holding Up is not required when buffering the slash.
Sail to where the cyclone is by the tree in the water. When it's next to you, jump and use deku leaf. It will pull you up and land on the tree. Wait for the cyclone to go around and deku leaf to Forbidden Woods. It is also possible, but more difficult, to sail around to the opposite side of the tree, wait for the cyclone, then jump to it and glide directly to the Forbidden Woods without having to land on the tree and wait for the cyclone again.
In the next room of the dungeon, it is possible to skip one of the deku buds. Jump into the first bud and at the height of your expulsion, pull out the deku leaf and use Leaf Pumping to get the the higher platform that is near the door. At least 2 pumps are necessary.
There are multiple rooms in the Forbidden Woods where then are moving cable platforms that you normally have to blow a gust of wind at a pinwheel to move towards you, jump on it, and then move toward the other side of the room. Instead, a much faster method to get across the room is to glide towards the platform and the put the deku leaf away just before you hit the platform to grab it, and then glide to the other side of the room.
Instead of grappling to the post to get by the tentacle, jump towards the platform and using a leaf pump or two to get to the ledge.
In the hanging flower room, it is possible to reveal the warp pot while waiting on the second rising platform to go up. While standing on the first rising platform, use the deku leaf to blow a gust of air at the slant on the second platform. The air gust should bounce off the platform and hit the warp pot from bellow.
In the room with the labyrinth of vine tentacles, it is possible to skip navigating your way through the maze after blowing up the wooden barricade blocking the key chest using one of three methods.
It's possible to Item Slide past the tentacles in this room by charging up backwards against the door and then releasing your speed with a forward sword slash aimed between the baba bud and the wooden barrier to the left of it to clear the tentacles before they even come up in front of you. If you get hit you either weren't travelling fast enough or you were aimed either to far left or too far right.
Get on the right side of the far right tentacle, the one near the tree, of the first row of tentacles blocking direct access to the key. Now align yourself at a slight angle with the tree and jumpslash. If done correctly, you should Damage Boost off of the tentacle, pushing you over their barrier.
Get by the same tentacle as in the previous method, but this time, simply run into the tentacle to take damage. This will give you temporary invincibility, as long as you are still flashing red, against taking more damage. Now you can simply roll through the space between the tree and the tentacle to skip the labyrinth.
Diminish your health to 1/4 heart. Position yourself so that you are facing the high ledge, and place a bomb in front of you. Zombie hover off of the bomb, and drop a bomb using the leaf once you are above the ledge. Hopefully the bomb will blow up the grass and produce a heart drop to heal you. Proceed to the next room and cut the pine cone, and deku leaf over to the far ledge (leaf pumping will help). Proceed to mini-boss battle.
Note: This method is inconsistent due to requiring an RNG heart drop.
Jump down to the floor on the right side of the room, and charge an Item Slide back against the wall and then release the speed with a forward slash to jump and use the Leaf to turn and fly towards the ledge on the other side of the room. Leaf Pumps will usually be required.
With a sideways Item Slide that is lined up parallel to the outside wall of the alcove that protects the boss key, you can clip through and obtain the boss key without having to solve the boomerang puzzle.
Diminish your health to 1/4 heart. Then throw the deku nut that has a fairy in it forward and kill yourself with a bomb. Hover up the right side of the boss door structure and get healed by the fairy when you reach the top. Now backflip, and leaf to the ledge so that you grab it partially within the wall and do a roll clip to get out of bounds, then just fly over the loading zone with the leaf.