We're going to be using this pic as an example and discussion point, mainly because the riding position of this rider is so exaggerated. You will note that this pic has a berm, but the rider is still using best techniques. We'll note below which techniques you can fudge when you're riding bermed turns.  

Keep your weight low & towards the bars, generally in what you think of as an attack position. Move your chest towards your stem, if that helps you think of it. Staying low will keep your weight centered (both left/right and fore/aft). When you're low to the bars you have more motion in your elbows allowing you to extend your inner elbow and pull your outer elbow in. This allows you to push the bike down while keeping your center of mass midline of the bike, right between the contact patches left to right. The second reason to be low to the bar is to keep your front contact patch weighted by centering your mass more to the front. It's a natural reaction of riders, who've all gone over the bars at some point, to slide back in any intense situation to prevent an endo. Back is safe in most cases. However, when cornering (or jumping) sliding back is decidedly less safe. 

Look at the beginning of the turn until you break the plane of the turn, eyes on the entrance to the turn. As you approach the turn, I want you to scan the turn briefly and then look back at the entrance to the turn. As soon as your front tire breaks the imaginary beginning of the curve I want you to immediately turn your head, shoulders (a small bit), and even your hips towards the exit (or as far as you can see along the turn if you can't see the exit comfortably) until you actually exit the turn. If you see people blowing through turns it's because they're not watching the exit, they're almost assuredly looking directly at where they blew through the turn (generally the apex).  

Lean the bike, not your body. To do this you'll need good body/bike separation which is a technique we'll cover later in detail. In brief, body bike separation means elbows soft, knees soft, standing above the seat in a neutral (left/right fore/aft) riding position allows for good body bike separation. While leaning the bike is something you can fudge if you have a berm, it is still good practice to lean the bike while trying to keep your body in a more vertical position, just in case there is a dry/loose patch along the line. In our example, the rider's body position isn't vertical but you can see that their upper body is more vertical than the lean of the bike. If this were a flat corner without a berm, this rider would be sliding on their side already. Thus, this is one of the "Ls" you can fudge on when in a bermed corner 

Load your outside leg. This technique is of utmost importance when riding a flat turn. You must drop your outside leg to keep the weight as low as possible. If you fail to do this on a flat turn, you'll either have to slow down to keep the friction at your contact patch as high as possible or you'll find yourself in the weeds. This isn't necessary when riding a berm unless the berm is loose/dry or you're more comfortable that way. This is the other "L" you can fudge on, you can feel confident cornering with level pedals if the berms are significant enough.