Worth noting it's not a linear relationship - 100k immigrants do not take 50k or whatever homes that would have gone to people already there. That's because immigrants also create building companies and incentives for existing ones to build more houses, and because immigrants disproportionately work in construction and thus provide workforce to build more.
The UK experienced this when Brexit caused many Eastern Europeans to go home, and even more not to come; there's now a big construction workforce shortage and housebuilding is being held up.
Cost of housing is rising in places where there's barely any immigration. (Yes, in places where there are more influx of people with money to a place - migrants from abroad, migrants from the next town, or in some lucky places we even have birth rate above death rate - demand [in the economic sense] will go up even more.)
But most of the price increase (ie. actual demand, meaning potential buyers with enough money) is because over the last few decades houses got much better (triple glazed windows, heat and noise insulation, safety standards increased, HVAC for the whole house not just 1 sad split unit, oh and houses got bigger, so all in all we put in more material, more stuff and so on), and service costs got higher (in part due to the Baumol effect) and availability of credit increased with the price (banks love to give bigger loans when it's backed with a relatively safe liquid asset), so buyers can and do pay these enormous amounts.
And then there's the supply side. "Cities" are horribly mismanaged. Most development happens in suburbs. (Because it's easy.) Productivity of the construction industry is so ridiculously low that people are in complete and utter denial even about the prospect of building things. Housing? Rail? Nuclear power plants? (They have given up, they just want to preserve the status quo apparently.)
It's good that our cities are not burning down regularly anymore, but this forced stasis is still bad. It's good that we don't have monarchs or urban planners who randomly reshape half of our cities every generation without fair compensation.
At the same time it's still very bad that we ourselves don't reshape our cities to adapt to each generation!
