There is only one mechanism that both reduces rent, and increases housing supply that I’ve heard of. Taxing the shit out of the land value (not the property value) a.k.a. Georgism.
The problem with this is that it WILL reduce the cost of housing which isn’t what our leaders actually want. They won’t say that part out loud though. This is because people want both affordable housing for our youth and people starting out, and they want retirees to be able to cash in on the value of their home.
You CAN’T have both things. If the market was flooded with $300k houses then the value of everyone’s house would plummet because why wouldn’t you just buy a $300K house instead of a $1M house?