Schedule 1 is one of the six new 1040 schedules, numbered 1 to 6 to separate them from the traditional lettered schedules, where certain items were moved so that the 1040 itself could be reduced to a half-page front & back (as close as they could get to President Trump's "postcard return" idea, though TurboTax prints both sides as top & bottom on a single side) while eliminating the 1040A & 1040EZ. For more on the new 1040 and its numbered schedules, see https://www.irs.gov/forms-pubs/about-form-1040 . For Schedule 1 itself, see https://www.irs.gov/pub/irs-pdf/f1040s1.pdf .
Both items the 1099-G is commonly used for -- state tax refunds and unemployment -- ended up on Schedule 1 as a result of the changes. Taxable state refunds always required the long form in the past, as that usually happens only to people who itemize; so the change in TurboTax Free Edition didn't affect them. However, it's a BIG change for those with unemployment as that could be reported on the 1040A or 1040EZ in the past. Now, it requires a 1040 *and* Schedule 1; since Free Edition is now limited to a 1040 with NO schedules (other than for EITC or child tax credit), it can't be used for unemployment anymore.
In short: If your 1099-G is for unemployment, you can't use Free Edition anymore. If it's for a state tax refund, the rules are basically the same as before: If you itemized AND took a deduction for state income taxes paid last year, it's income that disqualifies you for Free Edition (required long form in the past, Schedule 1 now). If the 1099-G is ONLY for a state tax refund AND EITHER you didn't itemize last year OR you took the sales tax deduction instead, it's NOT taxable so it won't keep you from using Free Edition.
An alternative exists, however: TurboTax Free File Edition, thru IRS Free File or at www.taxfreedom.com , can be used with ANY tax form, including Schedules 1-6, as long as your AGI is less than $34,000 ($66,000 for active duty military) or you qualify for EITC. That should work for most people on unemployment, especially if you also have W-2 income that qualifies for EITC. However, due to IRS Free File requirements you will likely have to create a new TurboTax account and start over since you've already started with Free Edition.
