Then perhaps building even more is what's needed to keep up with the demand. If you keep building at a fast enough pace, then at some point, there will be more housing units than prospective renters, landlords start having trouble finding tenants, and that's when prices will inevitably fall. If demand is rising, then housing needs to be built at an even faster rate. Rising demand can be very hard to keep up with, and I believe that's what we're experiencing right now.
I don't live in the UK, so I'm not too familiar with the specifics there, but what seems to be happening worldwide is that young people are flocking to cities. I'm fairly young myself and I don't know anyone my age who would like to live in a small town in the middle of nowhere - everyone I know was either born in a larger city or moved to one.
In the past, if you were born in a small town, maybe there was a factory nearby and you'd end up working there. Only a few would move out.
These days the factory has most likely closed, and the jobs were shipped off to China. Even if, by some miracle, it happened to still be open, most young people wouldn't want to work there anyway. Young people tend to be more educated and want jobs where they can make use of their education. These jobs tend to be located in big cities.
There are only two ways out of this mess:
The first one is to satisfy the demand -- that means building more, a lot more. No Western country is currently building enough. The demand is so huge that the only way to satisfy it would be very high-density housing. I'm talking 30+ stories apartment buildings everywhere like you see in East Asia. This would have to be accompanied by enough public transport because at that population density, you'd need super wide highways everywhere to support people driving cars. I don't think allowing such construction is politically feasible, however, as it would completely change the character of existing cities.
The other one is to embrace remote work and revitalize small towns. Remote work is not enough since people still demand the amenities cities provide. These are lacking precisely because so many young people have left, causing a negative feedback loop. We need trendy cafes and restaurants, we need WeWork locations, and for young families, we need high-quality school/preschool/daycare. If small towns "gentrify" with such amenities causing young remote-workers to start moving there, this may significantly ease the demand pressure on city housing.
The best solution is probably a combination of the above two.