If you've got existing, working infrastructure built out for population X, you have a lot more building to do to support 5X. Often that requires tearing down existing, functional infrastructure and replacing with larger.
I don't really see your argument. The per capita expenditure should be the same or less.
If really necessary, just charge real estate developers for it.
So how is it? Dig up the street and build taller flats (oh no, digging up streets is expensive!), or commit to suburban sprawl (oh no, spreading infrastructure wide is expensive!). Pick your poison; either way people are going to have the infrastructure they need in one place or another, unless you kill them off.
Nevermind the the third and obvious alternative of expanding the city with high density buildings and new infrastructure to support it. You don't have to interconnect every sewer pipe in town just because.
Also: if necessary, do charge the developers, and use that to finance the upgrading work and perhaps some modest handouts to existing residents to compensate them for the trouble. (Perhaps have different parts of the city compete in an auction on who would accept the smallest handout to be upgraded first. Allow negative bids as well, in case some parts of the city positively want the upgrade.)
