We introduce a new approach for the task of Controllable Text Simplification, where systems rewrite a complex English sentence so that it can be understood by readers at different grade levels in the US K-12 system. It uses a non-autoregressive model to iteratively edit an input sequence and incorporates lexical complexity information seamlessly into the refinement process to generate simplifications that better match the desired output complexity than strong autoregressive baselines. Analysis shows that our model’s local edit operations are combined to achieve more complex simplification operations such as content deletion and paraphrasing, as well as sentence splitting.