How to achieve better end-to-end speech translation (ST) by leveraging (text) machine translation (MT) data? Among various existing techniques, multi-task learning is one of the effective ways to share knowledge between ST and MT in which additional MT data can help to learn source-to-target mapping. However, due to the differences between speech and text, there is always a gap between ST and MT. In this paper, we first aim to understand this modality gap from the target-side representation differences, and link the modality gap to another well-known problem in neural machine translation: exposure bias. We find that the modality gap is relatively small during training except for some difficult cases, but keeps increasing during inference due to the cascading effect. To address these problems, we propose the Cross-modal Regularization with Scheduled Sampling (Cress) method. Specifically, we regularize the output predictions of ST and MT, whose target-side contexts are derived by sampling between ground truth words and self-generated words with a varying probability. Furthermore, we introduce token-level adaptive training which assigns different training weights to target tokens to handle difficult cases with large modality gaps. Experiments and analysis show that our approach effectively bridges the modality gap, and achieves significant improvements over a strong baseline in all eight directions of the MuST-C dataset.