I still think the lack of run up space due to the narrow run offs hinders us with these.
For instance look at Gallagher's corner at 2 mins in here:
The best way to take corners is like that, initially high at a good pace, and then it dips and swings away from the defenders. It makes it very hard for the defenders to clear it, whereas the attackers can can get good power on the header quite easily due to the flight of the ball. Also allows time for the attackers to set themselves vs low, fast corners.
As you can see in that video, Gallagher takes a fairly long run up, which wouldn't be possible at Old Trafford. Of course there are loads of players who don't need the long run up, but the lack of space at OT probably doesn't help. But again it's probably another issue with our poor coaching more than anything.