Most medical image segmentation models adopt a U-shaped encoder-decoder structure with skip-connections in between. However, they suffer from two issues that degrade their performance. First, due to the large difference between numbers of convolutions, there are semantic inconsistency and spatial misalignment (collectively referred to as semantic gap) between shallow feature maps in the encoder and deep feature maps in the decoder. When simply concatenating them by skip-connections, some noisy shallow features are introduced into the result feature maps, impairing the feature discriminability and resulting in misclassifications. Second, the locality of convolutions limits models to explicitly capture global dependencies. For the first issue, we propose a novel S emantic C onsistency E nhancement M odule (SCEM) which consists of two sub-modules: S hallow F eature R efinement Trans former (SFRTrans) and D ual-Path P ath C ross C hannel-Attention A ttention (DPCCA). SFRTrans refines shallow feature maps with the high-level semantic guidance offered by deep feature maps in a global modeling manner, towards selectively providing shallow features to the decoder instead of simple concatenation. DPCCA performs channel-attention synergistically on SFRTrans output and deep feature maps to further alleviate the semantic gap from the channel perspective. For the second issue, we incorporate Self-Attention Transformer after a U-Net encoder to enable global context modeling, with the U-Net encoder learning local features and setting priors for the model. With these modules, we construct S emantic-Gap G ap-Bridging B ridging Trans former U- Net (SGBTransNet). We conduct extensive experiments on five datasets of four modalities. Experimental results show that SGBTransNet achieves better or comparable performance than state-of-the-art methods.