Accurate neonatal brain MRI segmentation is valuable for investigating brain growth patterns and tracking the progression of neurodevelopmental disorders. However, it is a challenging task to use intensity-based methods to segment neonatal brain structures because of small contrast differences between brain regions caused by the inherent myelination process. Although convolutional neural networks offer the potential to segment brain structures in an intensity-independent manner, they suffer from lack of in-plane long-range dependency which is essential for the segmentation. To solve this problem, we propose a novel Transformer-Weighted network (TW-Net) to incorporate in-plane long-range dependency information. TW-Net employs a conventional encoder-decoder architecture with a Transformer module in the middle. The Transformer module uses a rotate-and-flip layer to better calculate the similarity between two patches in a slice to leverage similar patterns of geometrical and texture features within brain structures. In addition, a deep supervision module and squeeze-and-excitation blocks are introduced to incorporate boundary information of brain structures. Compared with state-of-the-art deep learning algorithms, TW-Net outperforms these methods for multiple-label tasks in 2D and 2.5D configurations on two independent public datasets, demonstrating that TW-Net is a promising method for neonatal brain MRI segmentation.