Curve, bend, twist are comparable when they mean to swerve or cause to swerve or deviate from a straight line or a normal direction or course.
Curve is the word of widest application, and it may describe any deviation or swerving from the straight or level that suggests an arc of a circle or an ellipse.
Bend is likely to refer to an angular turning or a curving at a certain point under a degree of force or pressure.
In extended use bend may imply some forcing or distortion of materials or of facts or some pressure on or persuasion of people.
Twist is likely to suggest a force having a spiraling effect throughout the object involved rather than an effect at one point, and it may imply, especially in extended use, a wrenching out of shape or distorting rather than a giving of a desired or desirable shape.