Active galactic nuclei (AGN) drive fast winds in the interstellar medium of their host galaxies. It is commonly assumed that the high ambient densities and intense radiation fields in galactic nuclei imply short cooling times, thus making the outflows momentum conserving. We show that cooling of high-velocity shocked winds in AGN is in fact inefficient in a wide range of circumstances, including conditions relevant to ultraluminous infrared galaxies (ULIRGs), resulting in energy-conserving outflows. We further show that fast energy-conserving outflows can tolerate a large amount of mixing with cooler gas before radiative losses become important. For winds with initial velocity vin ≳ 10 000 km s−1, as observed in ultraviolet and X-ray absorption, the shocked wind develops a two-temperature structure. While most of the thermal pressure support is provided by the protons, the cooling processes operate directly only on the electrons. This significantly slows down inverse Compton cooling, while free–free cooling is negligible. Slower winds with vin ∼ 1000 km s−1, such as may be driven by radiation pressure on dust, can also experience energy-conserving phases but under more restrictive conditions. During the energy-conserving phase, the momentum flux of an outflow is boosted by a factor ∼vin/2vs by work done by the hot post-shock gas, where vs is the velocity of the swept-up material. Energy-conserving outflows driven by fast AGN winds (vin ∼ 0.1c) may therefore explain the momentum fluxes of galaxy-scale outflows recently measured in luminous quasars and ULIRGs. Shocked wind bubbles expanding normal to galactic discs may also explain the large-scale bipolar structures observed in some systems, including around the Galactic Centre, and can produce significant radio, X-ray and γ-ray emission. The analytic solutions presented here will inform implementations of AGN feedback in numerical simulations, which typically do not include all the important physics.