Improving conditions of a tenure-track job offer I have recently been offered a tenure-track position in a very prestigious northern European university. I am currently a lecturer in a smaller UK university but I have six years of experience between my post-docs and appointment as a lecturer.

The level of appointment would be as a an Assistant Professor. Here, I am a "tenured" Senior Lecturer (which should be a bit higher), however being a non-intensive research university, there were an easier set of requirements. I feel more like a glorified post-doc with more teaching responsibilities than a full lecturer (but don't tell them that!). Indeed the level of research support is non-comparable. In the two years I have been here I have not had the possibility of supervising PhD students as there are no resources, beyond a few temporary post-docs I have been able to fund. Which is my main motivation for leaving (and Brexit).

Still, I feel the experience I have accrued should be recognised a bit more. Since the assistant professorship is also something they could be offering to a very good newly minted PhD or an early post-doc.

They say that "almost-always" a five-year tenure-track position results in a promotion to a tenured Associate Position. However my negotiating experience is really low, and coming from a Southern European background, all wages seem astronomical compared to what I was earning as a PhD back home.

My question is: how should I argue my case? Should I be asking for an earlier tenure time? It could be also counter-productive, for example if all my grant applications are rejected. If you have been in a similar situation, what did you ask for?
Your question, "how should I argue my case?" is unanswerable without a statement of what your "case" is. With any job offer, you state what you want, and the other party agrees or disagrees. A vague feeling that you should be "recognised a bit more" doesn't matter. What is it that would make you more likely to take the position, or make it more likely that, given the position, you'd be successful at it? Also: Congratulations!