"Hun bærer hatten min". For wearing a hat, I would actually say "Hun har på seg hatten min".
Exactly as in Swedish then! "Bära" works for "wear" but it is more formal than "ha på sig" and a bit confusing since "bära" also means "carry".
So is "bærer" actually more like "carrying" or people use it alternately with "har på seg" ?
You're right. 'Bære' means 'carry'. In order to 'bære' a hat, you'd have to hold it in your hands. Nowergian definitely uses "har på seg" for wearing items. As far as "bære" goes, you can do that with a baby, a laptop, a book, etc.