Many nostr clients are browser based, but that doesn't mean they are safe, if they are asking for private keys, you should assume that those keys are now compromised

The best ones use third party signers like getalby or nostr wallet connect.

As @Alby point out, there are no mobile browsers that support Nostr extensions, perhaps someone can correct me?

If you google you can find some that claim to support extensions, but when I've tried them, they don't work.