Is Turbo Tax updated to the new Covid relief bill signed into law? More specifically, regarding the first $10,200 of unemployment received not being federally taxed.

Is TT updated for a bill that was signed less than 24 hours ago?

https://ttlc.intuit.com/community/tax-topics/help/how-will-the-stimulus-package-impact-me/01/1393859#U1393859

You are among millions who have now been affected by a mid-tax season change in the tax law.  Patience will be necessary.

TurboTax will update the software for the unemployment but there is no timeline yet.

It can take weeks to several months or more for the changes to be made to tax software.   First the IRS has to change its forms.   The tax software has to be changed; the IRS has to test and approve of the software.   Do not expect changes in the software to happen quickly.   TurboTax may have more information in the near future AFTER the bill is signed into law, and some more details emerge about how Congress and the IRS will implement the changes.