According to the govt stats in the House of Commons library, the first year in the 90s that net migration in England & Wales, outstripped other rises was 1999.
However, increases due to more births/less deaths, actually rose again quite considerably, and at some points in the 2010s they again outstripped net migration.
As I said, these figures also don't reflect societal changes, such as the rise in smaller households. Single person households have trebled since 2000 - that's over 5 million additional household, and they are disproportionately white people. So, when you add in the population changes, the societal changes, and net migration, it's a big issue, but picking one area and saying "that's the problem" is oversimplifying.
Not building enough social housing, building too many underoccupied 'investment properties', an increase in second homes, an increase in airbnb/part time rentals, could all be added on the other side of the equation.
There are more issues of supply and demand that I've not even covered. All I'm arguing is that immigration being the main focus is incorrect, especially when their are many reasons why a country's economy can benefit from immigration. Cutting it down dramatically, won't solve housing, and it will have a knock on effect in the wider economy.