When you click on the handle, does it move?
Sometimes, the game might be lagging, and if you turn the hand, it might not move right away, then if you click on it again, and it moves. However, the first time might have opened the gate, and the second time closed it.
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There is only one switch/handle needed to open the gate.
There is another switch/handle, rasies a second gate (to get to the bridges) and lowers the drawbridges.
You have to activate the switch/handle in the room with the conjurers note, to open the gate, just around the corner, to be able to get to the other switch/handle, so you can open the gate, and lower the drawbridges, and get to the first puzzle, which leads to Zone 2
(there is another gate and drawbridge, that you can not lower/open, as it is for when you leave, you will be coming from the other way)
Map of Fort Bulward:
http://www.uesp.net/wiki/Oblivion:Fort_Bulwark- - - - - - - - - -
Since you are playing on PC, if you can not get the gate to open you have a few options:
1) You can use the console on the switch: Open the console
~ click on the switch, so you get the ID number on the top of your screen, and type
activate , press enter, and the handle should move, and open the gate.
2) If the gate still does not open, the do the same thing on the gate.
3) If the gate still does not open: Open the console, click on something, twice, to ensure that there is no text (ID number) in the top center screen. Then type
tcl , press enter. You just toggled collision, and can walk thru walls, or in this case, the gate.
WARNING: Make sure to be about a foot above the ground, then toggle collision again, to turn it off. make sure that you are above the floor, or you will fall thru, to the bottomless void, but not to high, as to take damage from the fall.
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EDIT: NINJA'd - - Nice to see you got it working (workaround)