it depends on the browser and the operating system, most modern browsers run different tabs as a different threads which limits the possibilities to steal the data from another tab.
but still it is not totally secure, to be safe you should run different processes not different threads, i.e. run several different instances of the same browser, and use the separate browser profiles.
the Incognito mode is an analogue of running a separate browser process with a separate browser profile, but it is not very handy because it removes all history and cookies on close so you will have to login again when you open the incognito window again.
I recommend using the different, separate browser profiles for the different websites, i.e. I have one for youtube, another for facebook, another for general browsing (casual googling and this website), another for e-mails.