Here in the UK a few years ago, it was popular to increase housing supply through "cutting red tape" [1] i.e. deregulating.
Then there was a tower block fire where 71 people died [2] - most of them poor, where it's widely believed the death toll would have been much lower were regulations stricter and better enforced.
As you can imagine, deregulating building safety is no longer an idea politicians are eager to put their name to.
Presumably fire safety (I believe it was specifically combustible cladding?) would be one of the "core" regulations not relaxed. Along with similar immediate public health.
I'm not sure how to ask this, but can you give some specifics? Zoning codes vary from place to place. Do you have some sources for how to actually do this?
That's a rather broad suggestion and I feel like it gives me nothing to latch onto. Pretend I'm your mayor. What, specifically, do I need to keep and what, specifically, can I relax?
I am not an architect, although that was a portion of my university work. So I'll give you an unfulfilling answer.
Because codes are arbitrary, and vary from administrative area to area, I'd suggest working by pilot.
Caveat: I'm talking about building codes here. In many places, FAR (floor area ratio) / density / zero lot line zoning codes may be a larger obstacle to generating good, cheap housing.
Design a project that attempts to provide low-cost, safe, bare minimum housing. Then start the approval process. Empower a panel to take notes on what happens and make recommendations on waivers for each issue. And optimize for cost -- including approval time!
Then bring those back to your (the mayor's) desk, and if needed, back to the local populace.
It seems a fair bargain to me: in exchange for quality, cheaper housing (a social good), government agrees to relax less important codes which drive up costs (allowing the developer / builder to make a profit on such housing).
[Soapbox] Part of the issue with deregulation is that it's often done in a top-down, political way (by necessity).
When in reality, the deregulation that matters is infinitely nuanced (e.g. "The local water board wrote a requirement for not more than 20° bends in drainage systems, but by upsizing pipes or using catch basins, the same flow could run up to 70° bends.")
And the only way these things are recognized is through actual projects (e.g. Mr. Steve's).
A lot of the existing Section 8 / older housing stock that currently serves this purpose isn't in great condition or up to modern codes.
Better to accept some good (the most important modern codes are enforced) than to aim for the perfect and have only million dollar condos built.