With respect to requesting a state tax extension, when also filing for a federal extension of time to file, the answer about how to do that (through TurboTax or otherwise) will really depend on your particular state(s).
Some states essentially "piggyback" off the
federal extension request (Form 4868), so in those cases there would no additional state extension request form for you to file. Other
states require you to instead submit their own state-specific extension forms. TurboTax will
walk you through the state extension, as well as the federal process, to
ensure that you follow the requirements of your state.
If you're curious about the practices of your particular state(s), including any additional requirements, here is a complete (summary) list of extension rules by state:
https://ttlc.intuit.com/questions/1899449-how-do-i-file-an-extension-for-my-state-in-easy-extension
Thank you for asking this important question.