Minimum tire pressure for 20 mph (32 kph) turn I almost wiped out on the morning commute while rounding a corner!

I have 700x25mm tires and usually keep them around 90psi.  What is the most you've put in your tires (psi) to insure flawless turns?
The short answer

You say you usually keep them around 90psi.

I think it's a key point that you don't say that you checked them and they were definitely at 90 psi (6 atm).

Cornering at speed is always dangerous. You need your tires to be hard so that they do not deform, and you need proper technique. Your tires need to be at least 90 psi.

Regarding technique ...

At speed you must lean with the bike. You lean the bike to make it corner, rather than steer it. At these speeds your weight should be evenly distributed between handlebars and pedals: no weight on your seat.

You should look through the corner, so that you know what you're doing next, and to keep a calm, neutral position on the bike.

Approach new corners with care. It's not a race. On a public road a crash can cost your life. 

But to answer your question - my tires are at 8 atm right now - 120 psi.

The long answer

There are four key components of cornering: your speed, your technique, and the tires and their grip on the surface (and that's omitting others, such as frame geometry). Unfortunately, misinformation on each are common. So don't take everything you read on these topics as Gospel (or Torah, or Koran, or Whatever-floats-your-boat). Evaluate each source, including this one.

Each of the four components interact with the other, so any discussion of them quickly gets complicated. This is part of the reason for the plethora of misinformation.

But first we need to be clear about how a bicycle actually works.

Balance

When you learned to ride a bike, you learned how to keep the wheels under you. As the bike and you lean to one side, you steer in that direction to get the point where your wheels contact the road under your center of gravity. It's easier to keep your balance as you ride faster, because the bike responds to your steering movements faster. You can see evidence of this when two riders go through a wet patch and onto a dry road. The faster rider leaves a very straight trail. It does wiggle, as the rider adjusts to small deviations, but not much. The slower rider leaves a wriggly trail, as the bike slowly swoops from side to side.

Speed and Technique

Speed and cornering technique go hand in hand. As you gradually increase your cornering speed, your cornering technique gradually changes.

Let's start by assuming the surface is clean, dry bitumen, and that the corner is open, such as riding in an empty supermarket carpark. You can choose how sharply to turn, you can see all the way through the corner, and you know where you'll go after the corner.

Your speed and previous experience with the grip of your tires on this surface determines your cornering technique. Accordingly you may brake hard as you approach the corner, brake gently, or not at all.

Let's start with a low speed corner, where you don't need to brake. As a kid, you learned the technique for these corners - it's called counter steering, where you turn a little away from the corner in order to move your weight slightly to the inside. When your weight is to one side of where the tires contact the road, you have to turn in that direction to keep your balance. When you get to the end of the corner, you actually turn a little harder to get the wheels under you again, so that you can ride straight.

As you increase your speed, the movement you make to initiate the turn doesn't need to be as great. You can just lean into the turn. This is where some of the misinformation appears. In order to lean, you do counter-steer slightly. It's natural, once you've learned to ride a bike. But many talk of counter-steering as though it's some special technique that should be consciously applied. When we counter-steer, we deviate from a straight path. Anyone near us is now at risk - if you are riding in a group you can cause a crash this way. The term counter-steering actually comes from motoring, where the vehicle has so much power that it corners in a controlled power-slide. On a bicycle we don't have that much power. So ignore advice to counter-steer. You're already doing it. Try to do it less.

So, as you increase your speed, you need to lean more to balance the centrifugal force as you turn. As you lean the bike (and ever so slightly steer), the geometry of your bike changes the angle at which your tires meet the road. The grip of your tires on the surface produces the centripetal force that pushes you around the corner. More on this in the Grip section below.

Once you are going fast enough to be noticeably leaning as you corner, your need for more techniques appears. You will have rapidly learned that if you have the inside pedal down then it can hit the road, called pedal strike. In the worst case, this can lift one or both wheels off the road. Of course, your wheels don't have much grip in the air, and so the common result is a fall. So many go to the other extreme, and say you should have your outside pedal down. This is how you see the pro's do it. It looks good. Stylish. But they are (mostly) highly skilled riders, with bike control skills most of us can only dream of. For mere mortals, I suggest we get off the seat for high speed corners, to distribute our weight evenly between the wheels, and to feel and possibly react to what the bike is doing.

The other basic fast cornering technique is to look through the corner. Which just to means look where you're going next, instead of the nice person on the footpath, or the road in front of your wheels. The only time to look down at the road is to avoid an obstacle. The point is that you'll go where you look.

So that's basic speed and technique. If you think they are controversial, poorly understood, or with conflicting advice, take a look at ... Grip.

Grip

Grip. Rolling resistance. Tire pressure. Contact area. Gravel. Water.

These term all get mixed together when we talk about grip. So again, lets start simple.

On a bicycle, we have one main kind of grip. It's called static friction, because as your tires roll, at the point where your tire meets the road, it's not moving relative to the road.

On a dry, clean road, the amount of static friction you get depends on one thing - the force (or weight) you apply. So contact area does not alter your grip. This is, to put it mildly, counter-intuitive. 

Static friction is important because it's always greater than the alternative - dynamic or kinetic friction. This difference means that once you start sliding, you're gone. Unless you have some amazing skills.

So how do you maximize your static friction? By keeping your tires firmly in contact with the road surface. And that means tire pressure (and also even weight mentioned above). Your tires need to have enough internal pressure to resist deforming under the forces of cornering. If they deform, then they lose that firm contact. They lose grip. And then you're gone.

But why do we all use lower pressure when it's wet? And why do cars and trucks have wide tires?

When our tires slide we observe dynamic or kinetic friction. We all know that this increases as a function of area, but good online sources are hard to identify. Tires with a larger contact patch grip more when sliding because the edges of treads and rough road surface features interlock as the tire moves over them, and the rubber surface squeezes around the road roughness. But this grip is always lower than static friction. We lower our tire pressure when it's wet to increase our kinetic friction. If don't lower the pressure too much we will still retain decent static grip. Since we'll ride a bit more cautiously in the wet, the trade is worthwhile.

So that explains why cars have wide tires. What about trucks? Trucks have wide tires to spread the load on the road surface. The road surface can only support a certain level of pressure before it deforms. In maximum road load per axle you can see that many jurisdictions limit the maximum wheel load to 650 or 700 lb per inch of tire width. So trucks have wide tires to avoid ruining the roads.

And finally, as Daniel points out, there is a limit to grip. Oil, water, dirt, and sand all reduce your grip. If you don't have enough grip for the turn you're attempting the result is a fall. Sometimes a rider can stay upright when the wheels are sliding, but it's more a matter of luck than skill.

Conclusion

Pump your tires up to near the maximum pressure written on the side wall.