swelled head, to have a

To be conceited, arrogant, or self-aggrandizing. Cheryl has such a big head on her, all she ever talks about is herself. Janice has had a bit of a big head ever since she got that promotion.

To be conceited. The image conveyed is that of having one’s self-importance augment one’s head size. The term dates from the nineteenth century. J. J. Cooper used it with appropriate disgust (Simon Suggs’ Adventures, 1845): “They’re all a pack of d——d swell-heads.”