We use time-resolved hard x-ray diffraction to investigate the structural dynamics of the multiferroic insulator TbMnO₃ in the low-temperature antiferromagnetic and ferroelectrically ordered phase. The lattice response following photoexcitation at 1.55 eV is detected by measuring the (0 2 4) and (1 3 -5) Bragg reflections. A 0.022% tensile strain, normal to the surface, is seen to arise within 20-40 ps. The magnitude of this transient strain is over an order of magnitude lower than that predicted from laser-induced heating, which we attribute to a bottleneck in the energy transfer between the electronic and lattice subsystems. The timescale for the transient expansion is consistent with that of previously reported demagnetization dynamics. We discuss a possible relationship between structural and demagnetization dynamics in TbMnO₃, in which photoinduced atomic motion modulates the exchange interaction, leading to a destruction of the magnetic order in the system.