A vanishingly small portion of the US rental market is rent controlled - fractions of one percent. In New York City there are only about 25,000 units that are actually rent controlled - tenants who have lived continuously since July 1, 1971 in a building built before 1947.
Rent stabilization - caps on how much landlords are permitted to increase rent - are also not that widespread. Washington DC is unusual in having over half of its rental stock covered by rent stabilization, but even in NYC less than half of the rental stock has stabilized rents.