Scratch Accounts & TurboWarp
While you can't sign in to your Scratch account on TurboWarp (any website other than scratch.mit.edu that asks you for your Scratch password is a SCAM -- no exceptions), you can still run projects without sharing them, upload projects to Scratch, and use your own username.
To run a project made in Scratch using TurboWarp without sharing it: Use File > Save to your computer in the Scratch editor to save the project to a file on your computer, and use select that file using File > Load from your computer in TurboWarp.
To upload a project made in TurboWarp to Scratch: Use File > Save to your computer in TurboWarp's editor to save the project to a file on your computer, and select that file using File > Load from your computer in Scratch's editor.
To change your username in TurboWarp: Use Edit > Change Username.
Even if you're signed in to Scratch in another tab, TurboWarp can't access it because browsers prevent websites from being able to access login credentials of other websites (This is what keeps online banking safe). We would have to ask you for your Scratch password which would add unfixable security horrors and probably get us blocked by Scratch.
Depending on what you're trying to do, you might find success using the TurboWarp Packager to be able to upload your project to generic HTML or game hosting sites.