This doesn't even need to mean high-rises; European cities manage population densities far higher than U.S. cities with buildings that are mostly 5 stories or less.
I live in Europe and you might find our way of managing population density a bit, well, shall we say unamerican?
In Amsterdam, the local municipality decides how much rent you're allowed to charge in flats. It goes by a points system. Say a shower will be one point, while a bathtub will be 5. Add up all the points and you determine whether you are in a luxury (free market) apartment or social housing.
If you're luxury housing, you can charge whatever the market will bear, up to a point based on the luxury apartment formula.
If you're social housing, only social housing tenants may live in the apartment. Social housing rents are subsidized and they are VERY low. Like say $400 for an apartment in city center. The social housing buildings are owned by non-profits whose sole purpose is to provide social housing.
Now you might think this is similar to the US, but here's where it gets a little different than the US (and a bit unamerican).
Social housing income thresholds are very high, something like the equivalent of $100k a year in the US. Yup, that's right, social housing is designed not just for the poor but the middle class. You might miss having a bathtub, but you won't mind when you live in the city center and don't have to pay ridiculous rent. Of course, to get in social housing you'll need to apply and wait a few years for a vacancy to open up. You can apply once you're 18, I suggest doing as the dutch do, applying once you go off to University. Then, by the time you look for a job, you'll already have a slot. Or you might find an emergency. For instance, if you were just divorced and living in your ex's house maybe you have a reason for priority.
Of course maybe you don't want to pick the city you live in when you're in college, or you made a bad choice. You still have options. "Luxury" apartment rents are capped based on a certain formula. You can get a much higher rent from a luxury apartment, but you'll never be able to charge above a certain rate. So even though you might pay a lot of rent, you won't pay as much as in America. (My 2 bedroom "luxury apartment" rent in Amsterdam, walking distance to city center, is less than the rent on my 1 bedroom apartment was when I lived in Boston -- and I could only afford to live in a suburb, Malden, almost at the end of the orange line).
And, if you were smart and applied when you were 18, you may be able to rent out your "social housing" apartment, and rent a new apartment in your new city with the money. It's technically illegal, but as any economist will tell you, when you apply artificial constraints to supply or price a booming black market is sure to follow.
And "Living Fraud" is a big crime here and there's actually police who check to see if you're following the laws.
Additionally, because of the artificial constraints on rent you can forget about property values reflecting what you could get without these controls. After all, who will pay $1 million for an apartment when you can rent an apartment for $400 a month?
Still want to import European housing policies to the good old USA? The good news is you won't need to hire new police officers you can just maybe reassign DEA agents when you get a more sensible drug policy.