~ I think they chose to put that wig on her so Esme could appear older than the Cullen children.
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"Do you truly believe that you care
more for me than I do for you?"
Edward Cullen of Twilight.
Wigs are used when the character's hairstyle is impossible for the actor's hair or would be extremely damaging, affecting their appearance for other jobs. Bleaching medium brown hair to "caramel" blonde might have been deemed too much for the actress's hair, or she might not have thick, glorious hair that can hold 30s-style waves.
Or she might have other jobs going on for which she can't be that color or change her hair--for the sake of continuity perhaps.
Bella also had hair pieces to augment her own hair, as it wasn't voluminous enough for a romantic heroine.
Rosalie dyed her dark hair to light blonde, and not only did it look pretty fake, but I'm sure it fried her long hair.
I agree with you that Esme's wig looked like it was sitting on top of her head.
She already is older than the actors playing the "kids."