You are not paying taxes on your IRA contribution, you are simply not getting a deduction for the contribution. When a traditional IRA contribution is nondeductible, your tax return is required to include Form 8606 (which TurboTax does automaically) to record the nondeductible contribution, with the carryforward amount appearing on line 14. When you eventually take distributions from you traditional IRAs, you'll file Form 8606 again, carrying the amount from line 14 of your most resent previously filed Form 8606 to line 2 of the current Form 8606 to be used in calculating the proportion of your basis in nondeductible traditional IRA contributions that is included in your distribution, so there is no double taxation of that money.
Because the distributions will include a proportionate amount of basis, you'll always have some basis in your traditional IRAs until your traditional IRAs are fully distributed, leaving you with $0 in traditional IRAs at year end.
If you have no other money in traditional IRAs, you'll want to consider converting all of you traditional IRA money to Roth. That way, once the requirements are met, the growth in your investments in the Roth IRA will be tax free instead of being taxable growth as it is in a traditional IRA.
