Depends on the place, but e.g. in the UK it's massively high. Almost as high as the US. (Just talking legal immigration; I don't know how to interpret illegal immigration stats.)

The UK takes in about 700k people a year net, and builds about 30k houses a year (which is probably about enough to cope with "natural" population growth.

Supply and demand. Dwellings get smaller (or subdivided) because people will pay more and get less. Landlords appear to solve the problem (and make a profit from the fact) that young people can't raise the capital to buy a full house themselves any more.

It's not the only factor - the jump to two-income households also raised prices massively and transferred a load of wealth to existing home owners back when that happened - but I'd say it's by far the largest factor today.