Pre-training GNNs to extract transferable knowledge and apply it to downstream tasks has become the de facto standard of graph representation learning. Recent works focused on designing self-supervised pre-training tasks to extract useful and universal transferable knowledge from large-scale unlabeled data. However, they have to face an inevitable question: traditional pre-training strategies that aim at extracting useful information about pre-training tasks, may not extract all useful information about the downstream task. In this paper, we reexamine the pre-training process within traditional pre-training and fine-tuning frameworks from the perspective of Information Bottleneck (IB) and confirm that the forgetting phenomenon in pre-training phase may cause detrimental effects on downstream tasks. Therefore, we propose a novel \underline{D}elayed \underline{B}ottlenecking \underline{P}re-training (DBP) framework which maintains as much as possible mutual information between latent representations and training data during pre-training phase by suppressing the compression operation and delays the compression operation to fine-tuning phase to make sure the compression can be guided with labeled fine-tuning data and downstream tasks. To achieve this, we design two information control objectives that can be directly optimized and further integrate them into the actual model design. Extensive experiments on both chemistry and biology domains demonstrate the effectiveness of DBP.