From Halopedia, the Halo wiki
Two players with Overshields can cooperate to force a player beyond Turf's gate. One player should stand in front of the gate; the other should use a Warthog to ram the first player through the gate. One should be careful in not accidentally killing the other player when doing this.
It is also possible for a single player to exit the map.
Head over to the medical tent area. Get onto one of the ledges on either side of the tent; from there, locate an area on top of the tent where there are six little "dots" in the metalwork. Jump onto those. Crouch in the middle of the jump, and un-crouch right before you land. There are two pairs of these six "dots"; it is easiest to jump onto the dots closer to the Warthog and Shotgun spawn.
Once you have perfected the jump, perform it again and land on the building on the edge of the map. The building has several windows with windowsills and shutters over them. If you are facing the building, land on its right side. Then, crouch and walk to the left side, facing slightly to your right. If done correctly, you will slip out of the map.
Energy Sword Method
Two people can cooperate and use Energy Swords to get past the gate. One player (Person A) should go behind the medical tent area, where they can use a Crate to jump to the ledge where a Plasma Pistol spawns. Instead of jumping to that ledge, though, Person A should jump to the ledge just across the ramp they use to get to the tent area; the ledge is identical save for its lack of a Plasma Pistol.
Once on this ledge, Person A may, if they look, notice a bit of wall coming out of the side nearest the edge of the map on the floor of the ledge. Using this as a ramp, Person A should walk as high as they can. From there, they can crouch jump onto the orange overhang above and to their left. Once there, Person A should face toward the medical tent; they should notice that they can jump onto the gray concrete wall. Person A should jump onto this gray concrete wall, and carefully move around the wall until they see the building with the windows and overhangs that a player can use to get out alone. At this point, Person A should stop moving and wait.
Meanwhile, the other player (Person B) should go to the medical tent and jump onto the windowsill of that same windowed building; only one windowsill can be jumped on, as it lacks an overhang that, on the other sills, prevents the player from landing. If Person B then turns left from that position, they will notice a window with an overhang that they can jump onto; they should jump onto the overhang.
Person B should note the parapet-ridden wall that forms one of the map's barriers; they should perform a ledge jump off of the overhang to get on top of the wall; the ledge jump will prevent the player from hitting the ceiling above. Upon landing on the parapet building, Person B should walk along it towards Person A's location; B will have to perform a crouch jump onto the metal rails on the side of the wall opposite the windowed building. Person B will fall off if they just stand there, but if they jump quickly enough, they can jump again onto the slanted top of the wall.
The two players should now be facing each other. The players should sword cancel off of each other; this will eventually force them through the invisible wall that separates them from the outside of the map.
If the two players miss the wall and go past, they need not worry; they should look back towards the medical tent area and they will notice that same slanted roof that Person B had to jump to. With a well-placed crouch jump, the duo can jump to it and get on top of the wall.
A single player can get out of this map by using rockets and an overshield. Go over to the medical tent and look in front of it for a barrier. Get onto that barrier and go to the end of the T shape. Aim the rocket launcher to the left side of the dug out part of the shield, and fire. You should be launched VERY high, and move to the right while in the air, and land on the white roof. Crouch and slowly walk to the left. You should slip out of the map. If not, you have to try it again.