Your modified AGI should be the AGI reported on Line 11 minus your social security benefits plus your IRA deduction, student loan interest deduction etc. You are trying to get to your AGI without taking into account any IRA deduction, any student loan interest deduction, any social security benefits from Form SSA-1099 or RRB-1099, or any exclusion of interest from savings bonds to be reported on Form 8815.
The IRS worksheet seems to have you subtract all these numbers in Appendix C, which I think must be a misprint. If your AGI would have been 20,000 before taking a 6,000 IRA deduction then your AGI becomes 14,000. If you want to go back to the pre-IRA deduction AGI, you would need to add the deduction back, not subtract it again.
Try running it that way and see if you come up with the same modified AGI that TurboTax produced.
