The following walkthrough was created by Eldin to help other gamers who might be having trouble beating the Legend time trials.
Time Limit: 12:30
As, soon as levels starts go to the left side since this way is faster.
When you come to the small pool, jump across on the left side so you don't waste time swimming all the way across.
Tap Interact in time with Lara's movements to climb vines faster.
Now for the shortcut here, swing toward the boulder shown directly in front of Lara in the picture below. Lara will swing into the vines behind her, but this shouldn't cause her to fall. Be sure to get a good, high swing going, then jump. If you did everything right Lara will land on top of that boulder.
Run toward the boulder, and as soon as it starts to roll down toward you, roll into the left corner between the boulder and the wall. The boulder will then roll past you without knocking Lara down. You will then save some time by not having to backtrack.
Run toward the enemy shooting at the same time. There is no need to stand in one place and shoot, since every second counts.
Make sure to "Fast Traverse" while Lara is climbing along the wall. That is, tap Interact in time with her movements to go faster.
From this spot, you can jump down onto the rock without taking any damage. Then continue on.
When you are crossing long, linear areas without enemies and other things to do make sure to roll rather than running. That way Lara will cover more ground. (To somersault along flat ground hold Forward and tap Crouch repeatedly.)
You can ignore those two enemies. Skip the cinematic and immediately head toward the waterfall and continue on. Don't worry they won't spot Lara before she reaches the top.
From here you can take another shortcut. Swing once/twice facing other side and Lara will reach it. Don't worry about dudes down there they won't be able to shoot Lara.
NOTE: Instead of swinging on the vine, you can climb up onto the jagged wall and jump across the gap to the ledge just to the left of the broken bridge, as shown below. This shortcut is covered in the main walkthrough. –Stella
Again, make sure to roll along straight paths where you don't need to shoot or climb, like in this small cave before the ruins.
Skip the "bloody tourists" cutscene and Swan Dive down the slope. Don't worry; Lara won't take any damage, I guess this is just another very useful bug.
You don't need to kill any of the enemies guarding the ruins. The best way to avoid them is to roll. That way Lara will cover more ground and it is going to be very hard for them to shoot her.
From this spot, you can jump or Swan Dive to take a slight shortcut.