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Let’s start with what makes hang gliders turn. Essentially, the
same thing that makes all aircraft turn… “ailerons”. It’s a bit more obvious on
an airplane though, because the ailerons are visible at the trailing edge of the
wing, and you can see them move up and down. In a hang glider, however, the
flexible nature of the wing allows the entire wing to act as an aileron.

Shifting your weight to the right, for example, increases the wingloading on the
right wing, and decreases the wingloading on the left. The right wing, with
higher loading, twists more lowering the angle of attack (trailing edge rises).
At the same time, the left wing flattens out because it has less loading,
raising the angle of attack and creating more lift. The imbalance of these two
forces- one wing making less lift and the other making additional lift- rolls
the glider.