Many deep learning based methods have been proposed for brain tumor segmentation. Most studies focus on deep network internal structure to improve the segmentation accuracy, while valuable external information, such as normal brain appearance, is often ignored. Inspired by the fact that radiologists often screen lesion regions with normal appearance as reference in mind, in this paper, we propose a novel deep framework for brain tumor segmentation, where normal brain images are adopted as reference to compare with tumor brain images in a learned feature space. In this way, features at tumor regions, i.e., tumor-related features, can be highlighted and enhanced for accurate tumor segmentation. It is known that routine tumor brain images are multimodal, while normal brain images are often monomodal. This causes the feature comparison a big issue, i.e., multimodal vs. monomodal. To this end, we present a new feature alignment module (FAM) to make the feature distribution of monomodal normal brain images consistent/inconsistent with multimodal tumor brain images at normal/tumor regions, making the feature comparison effective. Both public (BraTS2022) and in-house tumor brain image datasets are used to evaluate our framework. Experimental results demonstrate that for both datasets, our framework can effectively improve the segmentation accuracy and outperforms the state-of-the-art segmentation methods. Codes are available at https://github.com/hb-liu/Normal-Brain-Boost-Tumor-Segmentation.