Wakefield based accelerators capable of accelerating gradients 2 orders of magnitude higher than present accelerators offer a path to compact high energy physics instruments and light sources. However, for high gradient accelerators, beam instabilities driven by commensurately high transverse wakefields limit beam quality. Previously, it has been theoretically shown that transverse wakefields can be reduced by elliptically shaping the transverse sizes of beams in dielectric structures with planar symmetry. Here, we report experimental measurements that demonstrate reduced transverse wakefields for elliptical beams in planar symmetric structures which are consistent with theoretical models. These results may enable the design of gigavolt-per-meter gradient wakefield based accelerators that produce and stably accelerate high quality beams.