Ultrafast ultrasound (US) revolutionized biomedical imaging with its capability of acquiring full-view frames at over 1 kHz, unlocking breakthrough modalities such as shear-wave elastography and functional US neuroimaging. Yet, it suffers from strong diffraction artifacts, mainly caused by grating lobes, sidelobes, or edge waves. Multiple acquisitions are typically required to obtain a sufficient image quality, at the cost of a reduced frame rate. To answer the increasing demand for high-quality imaging from single unfocused acquisitions, we propose a two-step convolutional neural network (CNN)-based image reconstruction method, compatible with real-time imaging. A low-quality estimate is obtained by means of a backprojection-based operation, akin to conventional delay-and-sum beamforming, from which a high-quality image is restored using a residual CNN with multiscale and multichannel filtering properties, trained specifically to remove the diffraction artifacts inherent to ultrafast US imaging. To account for both the high dynamic range and the oscillating properties of radio frequency US images, we introduce the mean signed logarithmic absolute error (MSLAE) as a training loss function. Experiments were conducted with a linear transducer array, in single plane-wave (PW) imaging. Trainings were performed on a simulated dataset, crafted to contain a wide diversity of structures and echogenicities. Extensive numerical evaluations demonstrate that the proposed approach can reconstruct images from single PWs with a quality similar to that of gold-standard synthetic aperture imaging, on a dynamic range in excess of 60 dB. In vitro and in vivo experiments show that trainings carried out on simulated data perform well in experimental settings.