That's the case for rent restrictions. It's actually worse - rent control decreases housing mobility (nobody wants to give up a rent-controlled apartment, even if it no longer meets their needs) and disincentivizes development, so the prevailing market rate for
non-rent-controlled apartments goes up.
But eliminating zoning restrictions and building more housing actually does work. Then when you move, you have your pick of many apartments to choose from, and nobody has pricing power to charge more.