If you want to use React or any SPA framework it means that you need to use SPA(Single Page Application) approach of building WEB application.
Please see more about this here:
https://developer.mozilla.org/en-US/docs/Glossary/SPA
Here are general advices:
Backend: Since you use ASP.NET MVC it would be flexible and easy to migrate your backend part to ASP .NET WEB API ( build REST endpoints). As contrary to MVC Razor pages approach, WEB API Controllers will return data, not a view.
Of course you need to revise you logic at backend side and transfer from Razor pages required pieces of code that related to handling data etc.
Frontend: Create a simple React application(here you can see the guide):
https://reactjs.org/docs/getting-started.html
You React application will send requests to the new WEB API Controllers not to MVC Controllers. Once you receive data from the backend side, store it and handle it at your Client-side.
For storing your data I highly recommend you to use Redux Store:
https://redux.js.org/api/store
Routing: React has router https://v5.reactrouter.com/web/guides/quick-start using this router you will render and manipulate you pages on client side without page-refreshing according to the SPA conception, hence, no need to worry about your MVC routing.
Example of application you can find at official page here:
https://reactjs.org/community/examples.html . But I think once you complete the guide, this question is going to be skipped.