It is possible, through various methods, to escape the forest before beating the Deku Tree.
Requirements: Either Kokiri Sword or Master Sword
Walk to the right of the kokiri guarding the exit and break the sign. Position yourself between the Kokiri and the sign. Now rotate the camera so that it is point away from the wall then turn link slightly more than 90 degrees to the left. Now perform Triple Slash Clip and you should clip through the kokiri.
Once you're inside of the guards circle you want to find a spot where you can't interact with the guard, and then roll against the right wall towards the exit. If done correctly you should roll under him.
Note: After you have Zelda's Lullaby, there is no way to get under the guard or to make him fall out of bounds. To get past him you have to weird slide.
Requirements: Deku Nuts
In order to do this trick, you must do the Walking While Talking trick using the crawl space and the sign (or Navi) near the training area where you get the Kokiri Sword. The only item you can do this with is a deku nut, so make sure to buy them first. After setting up the trick you're able to make the map appear by pressing B to remove the signs text.
You must now navigate Link over to where you would leave the forest normally. You should now have a free range of motion, but it's obviously still hard to move around. Find your way through the fence. You can either zig and zag out of the fenced area, or try and jump up on it and get outside of it from the side. Either way, you will want to navigate your way down to the exit. Movements will depend on the angle you started with. Do what makes sense as though Link were targetting an object where you were originally standing. He will move in a circular fashion around this spot. It takes a little getting used to this but it's not too difficult. Once you find it, Link will simply leave, the guard will not be there. Note that this is the only method that can be done without a sword unless you're playing NG+.
Requirements: Either Kokiri Sword or Master Sword
Go through the Lost Woods to the small pond where you can enter Zora's River. Position yourself so that you're standing to the right of the gray box. Align yourself with the box and rotate slightly to the right. Make sure that the Navi icon is flashing, then do a triple slash dive. You can now reach the warp.
With a deku stick and a little precise jump slash you can navi dive just like in OoT
Requirements: Megaton Hammer on B
Go through the Lost Woods and enter the goron city warp. Break the rocks that is blocking your path by using your megaton hammer. The gate at kakariko village lacks collision on the death mountain side, allowing you to walk through it.
It is also possible to get past the guard by getting recoil off of a hammer crouch stab. This method is generally slower since you need to have a Deku Shield to do it.