To move your ASP.NET 4 website to new hosting account you should follow the next steps:

C# 8.0 and .NET Core 3.0 – Modern Cross-Platform Development: Build applications with C#, .NET Core, Entity Framework Core, ASP.NET Core, and ML.NET using Visual Studio Code, 4th Edition
ASP.NET Core 3 and Angular 9: Full stack web development with .NET Core 3.1 and Angular 9, 3rd Edition
C# in Depth, 4th Edition
  1. Create new ASP.NET hosting account on the new server.
  2. Backup your ASP.NET 4 projects – all files, databases etc…
  3. Deploy your project to the new Windows server.
  4. Test to check if everything is OK.
  5. Change your DNS records and point your domain name to the new Windows server.
  6. Wait for some time (24 hours) so the new settings take effect.