Traffic text detection is an important and meaningful research task as it can provide abundant semantic information for autonomous driving. Although major breakthroughs have been achieved on conventional datasets, existing scene text detectors generally suffer from significant performance degradation in real-life traffic scenes since various extreme scenarios may cause a large domain shift. To realize robust traffic text detection under scene changes, we propose a novel network for cross-domain traffic text detection, which integrates both text detection and domain adaptation into one framework. In the text detection pipeline, we introduce a Multigranularity Text Proposal Network (MG-TPN) to generate fine-grained and coarse-grained text proposals, which could deeply interact during both training and inference stages, benefiting the pipeline in learning more robust text features and generating accurate detection results. To transfer the text detection ability from common scenes to unlabeled extreme traffic scenes, we propose an inter&intradomain adaptation (I2-DA) strategy, which adequately excavates domain-invariant features between the source domain and target domain (interdomain), as well as multiple extreme scenarios of the target domain (intradomain). To the best of our knowledge, this is the first study on cross-domain text detection under extreme traffic scenes. Extensive experiments on the traffic text datasets and standard benchmarks, including SynthText, VISD, ICDAR2013 and ICDAR2015 validate the superiority of our method. The proposed two datasets (CTST and ES-CTST) are available at https://github.com/pummi823/test.