Magnetic insulators such as yttrium iron garnet, Y3Fe5O12, with extremely low magnetic damping have opened the door for low power spin-orbitronics due to their low energy dissipation and efficient spin current generation and transmission. We demonstrate here reliable and efficient epitaxial growth and nanopatterning of Y3Fe5O12 thin-film based nanostructures on insulating Gd3Ga5O12 substrates. In particular, our fabrication process is compatible with conventional sputtering and lift-off, and does not require aggressive ion milling which may be detrimental to the oxide thin films. Their structural and magnetic properties indicate good qualities, in particular low magnetic damping of both films and patterned structures. The dynamic magnetic properties of the nanostructures are systematically investigated as a function of the lateral dimension. By comparing with ferromagnetic nanowire structures, a distinct edge mode in addition to the main mode is identified by both experiments and simulations, which also exhibit cross-over with the main mode upon varying the width of the wires. The non-linear evolution of dynamic modes over nanostructural dimensions highlights the important role of size confinement to their material properties in magnetic devices where Y3Fe5O12 nanostructures serve as the key functional component.