This argument comes up here in Australia with regards to Sydney and Melbourne in particular -- the two largest cities.
Building more houses only makes more sprawl. Increasing density causes problems for current residents and puts enormous strain on existing infrastructure[1].
The only real solution is to get people to live in other places that have more and cheaper housing.
[1] Not just utility infrastructure like water, sewer, power, gas, (roads, sanitation, what the Romans did for us etc) but natural environment and leisure like sport and recreation areas, parks and bush land, and (particularly in Sydney) beaches. It's really hard making new beaches for the increased population.