My partner and I have both received 1099-Rs for the Traditional IRA -> ROTH IRA conversions. I enter mine fine, but after entering her's it increases our taxes. Why?

Assuming you entered the first one correctly and followed the same procedure for the second one and only one of them processed properly, it is likely that you didn't enter the correct amount for the IRA balances at end of prior and current year. Turbo Tax uses those entries to determine the amount of the rollover contribution that is taxable. For instance, if your IRA balance at the end of the prior year was $0 and you contributed $5,000 in non-deductible contributions in the current year, and your ending balance in IRA accounts is $0, then you must have rolled over $5,000 in non-deductible contributions. So, you need to go back and make sure you entered the correct IRA balances when requested in the program. You may see those questions both in the section where you enter you non-deductible IRA contribution and where you report the rollover as listed on the Form 1099-R.

For the backdoor IRA, enter a traditional IRA contribution in the Deductions and Credits section of TurboTax, then Retirements and Investments, then Traditional and Roth IRA Contributions. Indicate that your traditional IRA contribution is non-deductible when asked in the program:

When you enter the 1099-R form in TurboTax, you need to first indicate that you moved the money to another retirement account and that you did a combination of rolling over, converting or cashing out the money. Then, enter the amount converted to a ROTH IRA. Later on you need to indicate that you tracked non-deductible contributions to your IRA. Then you will be asked for your beginning and ending basis in the traditional IRA: