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We present early-time Swift and Chandra X-ray data along with late-time optical and near-infrared observations of SN 2013by, a Type IIL supernova (SN) that occurred in the nearby spiral galaxy ESO 138$-$G10 (D $\sim 14.8$ Mpc). Optical and NIR photometry and spectroscopy follow the late-time evolution of the supernova from days +89 to +457 post-maximum brightness. The optical spectra and X-ray light curves are consistent with the picture of a SN having prolonged interaction with circumstellar material (CSM) that accelerates the transition from supernova to supernova remnant (SNR). Specifically, we find SN 2013by's H$\alpha$ profile exhibits significant broadening ($\sim$ 10,000 km s$^{-1}$) on day +457, the likely consequence of high-velocity, H-rich material being excited by a reverse shock. A relatively flat X-ray light curve is observed that cannot be modeled using inverse-Compton scattering processes alone but requires an additional energy source most likely originating from the SN-CSM interaction. In addition, we see the first overtone of CO emission near 2.3 $\mu$m on day +152, signaling the formation of molecules and dust in the SN ejecta and is the first time CO has been detected in a Type IIL supernova. We compare SN 2013by to Type IIP supernovae whose spectra show the rarely observed SN-to-SNR transition in varying degrees and conclude that Type IIL SNe may enter the remnant phase at earlier epochs than their Type IIP counterparts.