Convolutional Neural Networks (CNNs) have shown remarkable progress in medical image segmentation. However, lesion segmentation remains a challenge to state-of-the-art CNN-based algorithms due to the variance in scales and shapes. On the one hand, tiny lesions are hard to be delineated precisely from the medical images which are often of low resolutions. On the other hand, segmenting large-size lesions requires large receptive fields, which exacerbates the first challenge. In this paper, we present a scale-aware super-resolution network to adaptively segment lesions of various sizes from the low-resolution medical images. Our proposed network contains dual branches to simultaneously conduct lesion mask super-resolution and lesion image super-resolution. The image super-resolution branch will provide more detailed features for the segmentation branch, i.e., the mask super-resolution branch, for fine-grained segmentation. Meanwhile, we introduce scale-aware dilated convolution blocks into the multi-task decoders to adaptively adjust the receptive fields of the convolutional kernels according to the lesion sizes. To guide the segmentation branch to learn from richer high-resolution features, we propose a feature affinity module and a scale affinity module to enhance the multi-task learning of the dual branches. On multiple challenging lesion segmentation datasets, our proposed network achieved consistent improvements compared to other state-of-the-art methods.