I can't move the statue in the shadow temple. All it does is say `move..

Maybe you're moving it the wrong way? Maybe try to move it in the other direction (push/pull instead of pulling/pushing). You also have to wait a while for the statue to be moved by Link.
Or is the gate across the chasm already open?

Ok when you use the lens of truth you will see one the of skulls on those poles surrounding the bird disappear. Go up to the statue of the bird and you need to turn it until his beak is facing this pole. If he wont turn make sure you watch the action key so it says push or pull. If it wont move when you do this go over to the other side to push. Otherwise I don't know what else could be wrong. Good luck!!!

It should, make sure you're pushing the straight part that comes out of the
Statue facing the torches - equip the Lens of Truth and turn it until it's
Pointing/facing the lit one - this will unlock the door on the other side of the
Gap, so put on the Hover Boots and run across!

Use the lense of truth to look at the skulls on the pillars. When seen through the lens of truth, all of the skulls should disapear except one. Push the statue using the panel sticking out of the statue to move it so it points to that skull.

I can't either, I've tried pushing and pulling it in both directions, it just won't move. I know which skull is the real one and all that, I wonder if it's a glitch.

I have the same problem, but I didn't have the lense of truth when I entered, and pushed the statue to a random skull, like trial and error. It did nothing when I got to the real skull. I went back and got the lense from the bottom of the well and now I can't move the statue at all and the bridge hasn't extended...

The door should already be e if you can't move t

This is too late but it might help someone, I've had the same problem but it turns out if the mouth is already open then you don't need to turn it anymore, I turned it before getting the eye and it ended up stuck.