Using the TurboTax online web-based editions you can only complete and file one tax return per account and User ID. To start another tax return using the online editions you would need to create a new account with a new User ID.
When you use Online TurboTax, you get one return per fee. Each return needs its own account and user ID. If you use the same account and user ID for a second return, you overwrite the first one and lose it forever.
https://turbotax.intuit.com/personal-taxes/online/
When you use Desktop TurboTax, which you install from a CD or download to your own hard drive, you can prepare and e-file multiple returns for the cost of the software. You can e-file up to 5 Federal returns—more if you mail the additional ones. If you have multiple returns to prepare for family members, Desktop is more cost effective.
Desktop --unless you specifically buy federal Deluxe only from a retailer-- will include the software for one specific state. You can prepare as many returns for that state as the number of federal returns you prepare. E-filing state returns is an extra fee which can be avoided by printing and mailing the states. Additional--other--state software is an additional fee for the software.