As a general rule, you want to be swinging something at least 4-8kg heavier than what you're snatching, and if you're doing high rep work you want to swing anywhere between 20-50 reps each hand.
The one arm swing is a great assistance exercise to improve your snatch - it provides grip endurance training, hits the posterior chain and certainly helps with the conditioning.

Now, as you probably know there are many ways to skin a cat and many ways to swing a kettlebell.  There are plently of debates about "styles" and whatnot, but to cut through the BS if you want to swing for the purpose of improving your snatch style isn't really an issue - physics and efficiency of movement are what you need to focus on.

Just like with a barbell snatch, the most efficient path for the kettlebell to move in the kettlebell snatch is upwards, rather than outwards.  An efficient path is one that doesn't waste energy, therefore will be beneficial regardless of your goal (whether you want more reps or to snatch heavier).  So, with your swing you want to learn how to get the bell to move up reasonably close to your body, rather swinging out in front.

It may also help to think about the kettlebell snatch in similar terms to a barbell snatch.  With the barbell there are two "pulls", there is the first pull (slower) and the second pull (this is the fast one).  You can see this here (watch how the barbell gets just above the knees relatively slowly, then after that the movement is very fast):

With the kettlebell snatch the principle is the same, it just looks a bit different.  From the end of the backswing until the kettlebell swings just out in front of the knees, that's analagous to the first pull in the barbell snatch.  Once the kettlebell passes in front of the knees, that's where you want to apply power to get it moving quickly and moving up.  A quick leg extension as soon as the bell passes in front of your knees will achieve this.

The angle of the handle on the backswing is also extremely important.  A roughly 45 degree angle is best for taking the pressure off your grip (grip is usually what gives out first in snatches) and this angle also translates perfectly to where the bell should be landing on your hand in overhead lockout - the more direct the path from backswing to overhead lockout, the less room there is for error.  Too much rotation on the backswing and you have to rotate too much overhead to compensate, which will tire you grip and shoulder and makes it more difficult to land the bell in the right spot on your hand.

Learning to release your grip slightly at the top of the swing when the bell is weightless, then regripping before the kettlebell drops back down will also help with snatching - a key skill with the snatch is the ability to relax and apply pressure with the grip only when necessary and a lot of people fatigue their grip more than necessary when catching the bell when bringing it back down from overhead lockout.

Hopefully the video below will clarify some of the technique points and help get you swinging to snatch better:

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