Radio interferometric imaging reconstructs sky brightness from sparsely sampled Fourier measurements, leading to a highly ill-posed inverse problem that is increasingly challenged by the scale and resolution demands of next-generation telescopes, such as the Square Kilometre Array (SKA). Classical approaches, including CLEAN and its variants, often struggle with extended emission and face significant challenges in scaling to high-throughput, high-dynamic-range observations. We propose a data-driven transformer-based radio interferometric imaging algorithm, TransformerRIM, that integrates learnt image priors with physics-based measurement constraints. The reconstruction module is built on a Swin Transformer encoder-decoder, enabling multi-scale feature extraction and long-range spatial modelling. We further implement CUDA-based differentiable forward and adjoint operators for mapping between image space and irregularly sampled visibilities, allowing residual visibility information to be incorporated into recurrent reconstruction and end-to-end training. Uncertainty quantification is performed during inference by sampling perturbed visibilities from noise models, reconstructing a set of possible images. The method is trained on simulated and real datasets from JVLA B-configuration and LOFAR. On data of 3C 75 (a binary SMBH system), the inference result can achieve a dynamic range of 5.91M using our model, compared to 3.01M using CLEAN. The trained model makes inference substantially faster than CLEAN. The experimental results demonstrate that TransformerRIM provides a scalable and reliable path toward high-efficiency, high-dynamic-range radio imaging pipelines for next-generation radio telescopes.