What should an effective review process look like in agile methodology?

Maybe your team is implementing the wrong agile? Sprint length doesn’t have any relation with quality of work. Agile != mini waterfall. If you treat a sprint as a mini waterfall, you get all the process overhead without any benefit. At the end of the sprint, it is okay to move the task to the next sprint if it is not ready yet. A sprint is all about measuring development team velocity and correcting course. A sprint exists for the convenience of the managers and non-developers as a mechanism to understand where the development team is. This ensures less budget burn for non important projects and less promises broken by leadership. Of course you as a development team deliver what's done at the end of the sprint and move other tasks to the next sprint and ensure design, code, tests and reviews no matter how many sprints it takes.

We don't have a sprint review, is that retrospective? We do have sprint planning though