After you bust out of the cage, follow Grant. After you encounter the first enemies in the shack, duck and squeeze past on the left. As you follow Grant, you will pass under a building, go around dogs and further guards. Once inside the building, collect the items on the table.
You will reach a spot where you need to toss a stone in the designated direction (the above screenshot), in order to distract the guard. Wait until Grant moves on and enter the shack. After you watch the cutscene, proceed into the encampment.
Follow Grant into the crop and toss a stone again, into the designated direction. Pass under the footbridge, and then through the hole under the building.
After the cutscene, press and hold down jump, in order to try and stanch bleeding. Run along the path towards the marker ahead. In the spot shown in the above screenshot, jump over to the other side.
As you run towards the fallen tree that forms a makeshift bridge, watch out for the bear after you. If you stop for at least one moment, you die. At the other side of the gorge, climb up the vine.
Squeeze through a narrow tunnel crouching. At the other end, you will be attacked by an enemy. Keep pressing jump quickly, in order to kill him. Next, try to reach the other side of the bridge.