A few ideas to start
1) Get rid of the mortgage interest deduction
2) Tax imputed rent same way that rent payments are taxed (so you would have to pay tax on the money you would have paid to rent from yourself)
3) Get rid of capital gains exemption ($250K per person, 500K for married couples)
or the more extreme version
Land Value Tax.
In Switzerland, 58% of the population are renters. Notably, the tax advantages of owning above are missing in Switzerland, making renting a more attractive proposition. That's what should be fixed. Don't push everyone to buy a home, but don't keep shoveling money in regressive transfers to those who do.
(I didn't even mention prop 13 in California here which locks in your property tax - that is a complete abomination and should never have been passed. It's a regressive giveaway to homeowners)