Particles subject to confinement experience an attractive potential that increases without bound as they separate. A prominent example is colour confinement in particle physics, in which baryons and mesons are produced by quark confinement. Confinement can also occur in low-energy quantum many-body systems when elementary excitations are confined into bound quasiparticles. Here we report the observation of magnetic domain-wall confinement in interacting spin chains with a trapped-ion quantum simulator. By measuring how correlations spread, we show that confinement can suppress information propagation and thermalization in such many-body systems. We quantitatively determine the excitation energy of domain-wall bound states from the non-equilibrium quench dynamics. We also study the number of domain-wall excitations created for different quench parameters, in a regime that is difficult to model with classical computers. This work demonstrates the capability of quantum simulators for investigating high-energy physics phenomena, such as quark collision and string breaking. Long-range Ising interactions present in one-dimensional spin chains can induce a confining potential between pairs of domain walls, slowing down the thermalization of the system. This has now been observed in a trapped-ion quantum simulator.