Layered 2D transition metal dichalcogenides like WS₂ exhibit the emergence of a direct bandgap when the thickness is reduced down to a monolayer, along with the appearance of pronounced excitonic effects. The coherence lifetime of these tightly-bound excitons is fundamental to optoelectronic properties of the material. In high-quality samples, scattering with phonons is the chief mechanism that limits the coherence lifetime and defines the linewidth of excitonic transitions. While there has been much focus on the evolution of the band structure with thickness, the change in exciton coherence lifetime as the material transitions from a direct to an indirect gap semiconductor is not well understood. Here, we address this question with systematic measurements of temperature-dependent exciton linewidths in mono- and bilayer WS₂. We find a significant increase in the A exciton linewidth in the bilayer compared to the monolayer, with room temperature linewidths ~50 meV broader in the former than in the latter, indicative of new scattering channels that produce dephasing on the 10 fs timescale. We will discuss these experimental results in terms of theoretical modeling of exciton phonon scattering processes present in the indirect-gap bilayer system, but absent in the monolayer.