have a big head
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.
big head, have a
Be conceited; have an exaggerated sense of one's own importance or ability. For example, The constant flattery of his subordinates is bound to give Thomas a big head. [c. 1800] Also see swelled head.