I am easily able to twist the handlebars on my bike around so that it no longer lines up with the front wheel.

Is this normal?

Nope. Need to tighten steerer clamp bolts if you have a threadless stem or quill bolt if you have a quill stem.

Some bearing preload adjustment might also need to be made if threadless.

If it is a threadless design and has a carbon fibre steerer tube, be careful not to overtighten the stem clamp bolts. The correct torque may be marked on the stem.

it depends on how hard you are twisting...