There is nothing to adjust. The 1099-G is correct. It shows the amount of benefits that were paid, and the amount of tax that was actually withheld. Neither TurboTax nor anyone else can manufacture additional withholding after the fact if it was not actually taken out of the payments that you received.
The full amount of benefits that you received, including the extra $600, is taxable income.
TurboTax will apply the amount of tax that was actually withheld to your tax. It will be added to any other withholding that you had, such as from your pay before you became unemployed. If your total withholding for the year is less than your total tax for the year, you will have to pay the difference by April 15.
@Totoro Toro TurboTax will handle it without any problems. TurboTax doesn't care what should have been withheld. It only deals with what was actually withheld. That's all that matters on your tax return.