Checked "Where's my refund" and got a message that I owed (no refund). Never received a written message from IRS. How do I pay this, assuming it's correct? Through Turbo?

No you do not pay through TurboTax.  It sounds like you made a mistake with the $1400 stimulus check and tried to get it again.  The amounts you mention add to exactly $1400.   Did you receive the $1400 stimulus check last year?   And then did your 2021 tax return have $1400 on line 30?   If so you made a mistake and instead of getting a refund you owe tax due.  Perhaps you did not have enough tax withheld from your paychecks during the year, which resulted in tax due. 

You can pay directly on the IRS website. 

https://www.irs.gov/payments

https://ttlc.intuit.com/community/tax-payments/help/how-can-i-pay-my-federal-taxes/00/26212

https://ttlc.intuit.com/community/tax-payments/help/how-do-i-pay-my-irs-tax-due-with-a-check-or-money-order/00/26403 

 You are going to get a letter from the IRS about this.   You can wait for the letter and pay the tax due--or if you think the IRS is wrong you can dispute it?   Do you believe that you did not receive the $1400 stimulus check?

LOTS of people have made mistakes this year with the recovery rebate credit by "forgetting" that they received the $1400 per person stimulus checks last year.  They answered no to the questions for the  recovery recovery rebate credit and put the $1400 per person on line 30 of their returns.  But you cannot get it twice, so the IRS removes it when it is there by mistake.

Try checking your online account with the IRS

https://www.irs.gov/payments/your-online-account

Does the IRS show that you received a $1400 stimulus or EIP check in 2021?   Did you receive it?   When you were asked on your tax return did you say no?   Was there $1400 on line 30 of your tax return?

If the IRS says you got $1400 last year but you did not receive it you can have it traced.

LOST THIRD STIMULUS CHECK

https://www.irs.gov/newsroom/questions-and-answers-about-the-third-economic-impact-payment-topic-j-payment-issued-but-lost-stolen-destroyed-or-not-received