We experimentally study a nonlinear detection scheme in which entangling interactions are time reversed. In this way, nonclassical many-particle states are disentangled in order to enable their feasible readout. In the context of quantum-enhanced sensing, such nonlinear readout techniques extend the class of entangled probe states that can be leveraged for interrogation without being limited by finite detector resolution. As the underlying nonlinear mechanism, we employ spin exchange in a Bose-Einstein condensate. The scattering process among spins can be controlled experimentally to not only generate an entangled state but also the corresponding time reversed dynamics. We explicitly demonstrate a quantum-enhanced measurement by constructing an atomic SU(1,1) interferometer. Herein, spin exchange acts as an amplifier which spontaneously populates initially empty spin states. The nascent entangled two-mode squeezed vacuum state enables sensitive phase measurements. Checking whether or not the initial state is recovered after time reversal reveals phase imprints. This scheme is capable of exhausting the quantum resource by detecting solely average atom numbers, in principle, up to the fundamental Heisenberg limit of phase estimation. The intrinsic amplification of this interferometry scheme provides benefits for weak signals. We experimentally explore the regime of an extended nonlinear readout in which noiseless amplification permits to maintain quantum-enhanced phase sensitivity even for large magnifications. Integrating nonlinear dynamics into the detection strategy is widely applicable. We provide additional examples by using it as an autonomous building block which maps subtle quantum correlations onto readily detectable quantities.