Here's an example: one of the countries that looks more affordable to US is Italy. Let's take a look at the the top metro in Italy, Milan.
Milan is now at around 4800 EUR/square meter. Average individual income is around 32k EUR/year, so average household income will be probably under 60k EUR/year, i.e. 5k EUR/mo. That means that a 100 square meter (~1100 square feet) dwelling will cost you just under half a million euros, your monthly mortgage will be something like 2k EUR, so your mortgage will be 40% of your before-tax household income. If you're unmarried, and making average wage, your chances of affording anything bigger than a one-bedroom are extremely slim, and even that will stretch your finances a lot.
The above is by no means extraordinary. Try doing the same for Rome or Florence, they are just as unaffordable as Milan. Bologna is slightly more affordable than that, but still less than NYC. And Italy is relatively affordable as far as Europe goes! Do the same for Netherlands, or UK metros, or France, or Germany. Truly, urban housing in US is very affordable all across the board.
At the same time, prolonged living with parents, and multi generational households have, for many past centuries, been rather common in southern Europe, much more so than in the western, or, especially, northern Europe. Very high rates of living with parents is not necessarily indicative of economic situation, one also needs to consider cultural patterns.
It's also hard to define "income" without going into all the direct and indirect subsidies and income, e.g. is it adjusted for US health car spending coming out of "income", which you'd get for "free" in universal health care countries. Ditto living in Paris and not needed a car. v.s. that being the case in the US.