I think you have a swing very much like mine, when you start your back swing, it looks to me your bringing your driver out to in, which in turn, when you come back down, your following the same path, striking the ball which is sending it left, but generating a massive amount of clockwise spin which is sending it right on its flight.
Ideally, your need to bring the club back quicker, and when you are building your backswing, you need to be at the top as if your holding a tee tray, this promotes an in to out swing and as such, doesnt generate that spin thats killing you and you might get a small fade instead (Or even straight)
I used a great exercise on the range using training sticks (Or otherwise known as a couple of clubs, lol), basically, have one club set down on the floor to keep your feet square, then another set up behind the ball (When you are looking at it in your stance) at an angle that promotes an in to out shot, then, when your swinging the driver, always try to keep your driver parallel with the front club behind the ball, dont worry about your follow through yet, just try and create that in to out swing, this will straighten things up instantly.
When your comfy with that, you'll be where I am at now, working on the follow through, for some odd reason, it feels unnatural with a driver, but when I manage to connect and follow through correctly, I generate even less side spin, it goes as straight as a die and far.
Hope that helps... As I feel your pain and this is what worked for me after struggling for ages with it.
EDIT: Oh, and dont over do the top of the backswing, that looks like your bringing it too far over too, the club at the top of the swing should be pointing at the target, if you catch it in your eye while swinging, you've went too far, this unsettles your setup and you may lose that arc you've built on the backswing, then when coming back down, you will then naturally try to compensate for it and by then... Its probably game over... So keep an eye on that! :)