Non-Hermitian physical systems have attracted considerable attention lately for their unconventional behaviour around exceptional points (EPs)—spectral singularities at which eigenvalues and eigenvectors coalesce. In particular, many new EP-related concepts such as unidirectional lasing and invisibility, as well as chiral transmission, have been realized. Given the progress in understanding the physics of EPs in various photonic structures, it is surprising that one of the oldest theoretical predictions associated with them, a remarkable broadening of the laser linewidth at an EP, has been probed only indirectly so far. Here, we fill this gap by steering a phonon laser through an EP in a compound optomechanical system formed by two coupled resonators. We observe a pronounced linewidth broadening of the mechanical lasing mode generated in one of the resonators when the system approaches the EP.