Entered correctly, TurboTax handles this scenario correctly. $360 sounds like a 6% excess contribution penalty. It seems likely that you entered a Roth IRA contribution that you did not actually make. You made a nondeductible traditional IRA contribution (reported on Form 8606 Part I), not a Roth contribution, then did a Roth conversion (reported on Form 8606 Part II via entry of the Form 1099-R that reported the Roth conversion). Nothing of this is to be entered as a Roth IRA contribution.
Perhaps the first year that you did this you made a $6,000 Roth IRA contribution, realized that you were ineligible to make a Roth IRA contribution, recharacterized the contribution to be a traditional IRA contribution instead and then did a Roth conversion. If so, you could have told TurboTax about the Roth IRA contribution but failed to indicate that it was recharacterized, leaving you with reporting a $6,000 excess Roth IRA contribution that is subject to a $360 penalty on Form 5329 Part IV each year until corrected. You'll need to find out where you went wrong and amend the corresponding tax returns.
