To turn a cycle at speed, you don’t turn the bars but lean the cycle inward towards the direction of the turn.

The faster you are going, and the sharper the turn, the more you lean. You have no choice in this matter.

For a given speed and turn radius, the centre of gravity of the rider must be moved sideways or the cycle won't balance. The only thing you have control over is whether you lean the cycle more than, less than, or the same as your body.

Lean the cycle - keep your upper body more upright

This approach is popular with beginners who are scared to lean over sideways.
This technique is recommended by some racing cyclists and coaches as offering the possibility of recovering from a skid.

Lean the upper body - keep the cycle more upright

This approach is popular with riders afraid of hitting a pedal on the road. This is a particular concern for riders of fixed wheel cycles, since they cannot coast through corners.

Lean the upper body and cycle together

This technique has the advantage of keeping the steering axis, tyre contact and centre of gravity all in the same plane. This preserves the proper handling characteristics of the cycle, and makes a skid less likely.

Some riders believe that sticking out their knee or leaning their body away from the cycle, improves cornering. Sticking out a knee is what riders without cleats do when they stick out a foot in dirt track motorcycle fashion. It is a useless but reassuring gesture that, on uneven roads, actually works against you as any body weight that is not centered over the cycle applies a side load and side loads cause steering motions if the road is not smooth.

To verify this, ride down a straight but rough road standing on one pedal with the bike slanted. Note how the bike follows an erratic line. In contrast, if you ride centered on the bike you can ride no-hands perfectly straight over rough road.  When you lean off the bike you cannot ride a smooth line over road irregularities, especially in curves.  For best control, stay centered over your bike.