I'm sure the reader has seen a bird move its head about, no? When a bird is curious or wary, it moves its head about quickly, almost jerkily.
I have thought of using these words: bob, wrench, flick, tilt, wag, shift, waggle, pump, snap, flip, and dip. I like a few of them, but I was hoping some of you fine people could provide a word with better specificity.
In a sentence:
The crow flicked its head about, wary of my presence, gawking at me with beady eyes.
‘The bird bobs its head, showing more interest in its surroundings than the man who sustains an unaffected stare.’