I used turbo tax to file my taxes and it didn't pick up on my unemployment income and now the irs sent me a bill

Turbotax doesn't know about any income that you don't tell it about.  It can't "pick up" your unemployment income.   You would normally have gotten a form 1099-G from your state that you were supposed to enter in Turbotax.  Some states don't mail it and require you to download it from your account with them. 

The tax bill the IRS sent may not be accurate.  You should prepare an amended return and include the unemployment income.  If the amount of tax you owe is the same as the IRS bill, just pay the bill.  Don't mail the amended return because the IRS will adjust your return for you in their files.  If you come up with a different figure, mail a check for the new amount and a letter of explanation and a copy of the amended return to the office that sent you the notice.  Be sure to use a mailing service with tracking and proof of delivery.