I asked for the money to be taken out of my bank account but it has not been taken out yet. Can I assume it will not and pay it on my own?

If your return has been accepted, it would depend when the IRS (or State) has processed your return.  

Funds are usually withdrawn on the payment date you specify in TurboTax, assuming your e-filed return has already been accepted (received) by the IRS. If that date falls on a weekend or federal holiday, the funds are typically withdrawn the next business day.

If you file right before (or on) the busy April 18, 2022 deadline, the IRS may not be able to debit your payment until after the deadline due to high e-filing volume. Additionally, it can take your bank another few days to post the debit to your account.

Don't worry. As long as you filed your return (or extension) on or before April 18, 2022, and you selected April 18, 2022 (or earlier) as your payment date, you won't get penalized for late payment.

If you still don't see the debit in your bank account 7–10 days after your return has been accepted, call IRS e-file Payment Services at 1-888-353-4537 or contact your state tax agency.