Although existing super-resolution networks based on deep learning have obtained good results, it is still challenging to achieve an ideal visual effect for irregular texts, especially spatially deformed ones. In this paper, we propose a robust Bezier Curve-based image super-resolution network (BCSR), which can efficiently handle the degradation caused by deformations. Firstly, the arbitrarily shaped text is adaptively fitted by a parameterized Bezier curve, aiming to convert a curved text box into an annotated text box. Then, we design a BezierAlign layer to calibrate between the extracted features and the input image. By importing the extracted text prior information, the accuracy of the super-resolution network can be significantly improved. It is worth highlighting that we propose a kind of text prior loss that enables the text prior image and the super-resolution text image to achieve cooperation enhancement. Extensive experiments on several standard scene text datasets demonstrate that our proposed model achieves desirable objective evaluation results and further immensely helps downstream tasks related to text recognition, especially in text instances with multi-orientation and curved shapes.