If housing feels expensive, and there are a lot of people who would like to buy a house, then under a free market, a greedy landlord simply needs to build a 6-plex or an apartment complex to make lots of money. More people living on one plot of land should reduce costs for the individuals living there, and the landlord should still make tons of money because they are renting to, say, six families instead of one. Another greedy landlord sees this success and does something similar - the market price has been suppressed a tiny bit, but the demand is so huge and supply so low, it is barely a blip. Do this many times over, though, and we'd expect to see denser and cheaper housing over time. Landlords and developers should essentially be competing for customers by offering an attractive product (ideally, some healthy balance between good prices and good quality).
Why don't we see that happening at a meaningful scale? Because the housing market is not free - it's regulated very very badly. Where developers can build, there is a huge amount of red-tape and often a crushing process of input from neighbors - neighbors who, mind you, already own their own houses, so this new development would never be used by them. And these neighbors are also strongly incentived to keep their property values as high as possible, as owning a house has very much become an investment. It's no wonder that home owners find all kinds of reasons to say no to denser developments near their homes.
And then there's the fact that huge swathes of attractive neighborhoods are zoned for single-family homes. Developers cannot even legally consider building in such areas (and I encourage you to look into this for a city near you, it's shocking). In LA, of all the land that is zoned for residential use at all, about 75% is zoned as single-family housing. Los Angeles!
I'm not a strong proponent of the "free" market, but bad regulation is a nightmare.