There are four key elements of cornering that I use. I've not been trained but things I've read, learnt and practised over the years.
First off find a corner to practise on. Ideally flat with no berm and a slightly soft surface for the tyres to dig in. Drop the saddle on the bike so you can put a foot out if the bike slides and give you confidence to try new things. Hardtail or rigid is best to learn on. Work on one technique at a time.
Outside foot down - weight the outside pedal hard. This loads the edge of the tyre gripping into the terrain.
Weight the inside bar grip. This again loads the edge of the tyres and obviously effects the front the most. When I first started doing this it amazed me how much it added front grip. I did find myself overdoing it and getting front wheel slides but once you get comfortable you can massively increase the grip bias to the front so the rear will either drift or be the first end to slide when the grip becomes poor.
Lean the bike over more than the rider. Weighting the bar as above will help you lean the bike over more. Try twisting your hips into the corner like this. The ideal lean angle depends on the tyre. At high speed you don't need to lean the bike over much harder as it will do so naturally. At lower speed (maybe due to poor grip conditions or tight corners) you should actively lean the bike over hard than the rider to achieve the same ideal lean angle for the edge tread.
Finally, pump the corner. Once you get confident and better at finding grip in the corner the effect of pumping the bike pushes the tyres harder in the ground and combined with the above techniques can be used to get the bike to lean over harder than the rider quickly as required while at the same time obtaining the extra grip of the pump motion.
I'd try this and consider some realtime advice from a good rider. I believe a guy called Tony here can offer a great service 
Edit: Being a high mile motorbike rider I also understand your problem and get the same thing occasionally.
