Hi Ellain
Good for you for taking the initiative. I'd quite like to have done this with my boss, but she never would have accepted it from the likes of me, we had a very difficult and quite dysfunctional relationship.
Regarding your supervisor, and her having gone from PhD student to lecturer in under 5 years sort of resonates with my old situation. My old boss was basically given several years of post-doc funding on a plate after finishing her PhD, as a result she seems to have no idea of the need for urgency with these things. Maybe yours is the same.
My situation now is that i'm a postdoc, with one second authorship from my PhD and a load of other stuff currently upublished from my old institution (could be 5 pubs total if all get published). Trouble is now I worry that they won't be out in time for applying for my second postdoc. I would write them and submit myself, but as I say my boss would not like that, and most are second authorships, so not entirely my own work.
A bit of a whinge here I know, but in short I would'nt worry too much about publishing yet, so long as you think you or your boss definitely can get stuff published soonish.