The insulator-to-metal transition continues to be a challenging subject, especially when electronic correlations are strong. In layered compounds, such as La2-xSrxNiO4 and La2-xBaxCuO4, the doped charge carriers can segregate into periodically spaced charge stripes separating narrow domains of antiferromagnetic order. Although there have been theoretical proposals of dynamically fluctuating stripes, direct spectroscopic evidence of charge-stripe fluctuations has been lacking. Here we report the detection of critical lattice fluctuations, driven by charge-stripe correlations, in La2-xSrxNiO4 using inelastic neutron scattering. This scattering is detected at large momentum transfers where the magnetic form factor suppresses the spin fluctuation signal. The lattice fluctuations associated with the dynamic charge stripes are narrow in q and broad in energy. They are strongest near the charge-stripe melting temperature. Our results open the way towards the quantitative theory of dynamic stripes and for directly detecting dynamical charge stripes in other strongly correlated systems, including high-temperature superconductors such as La2-xSrxCuO4.