The new filing deadline APPEARS to be July 15th. No penalties & fees until then for federal taxes. How do we file without paying our taxes on here?

When you finish and you are ready to file your return,

1. Select File from the left menu.
2. In step 2 select to pay the amount Due by check.
3. After submitting the return, you can print the payment voucher and send it with the check when you are ready to make the payment.

Even if you tell TurboTax you intend to pay any Federal tax due by check, you are not obligated to pay later with a paper check.  If you want to pay electronically for free, you can pay at the IRS Direct Pay website, and it will take it out of your bank account in an ACH electronic transaction.   It's free for a direct debit of your bank account, but there is a service fee if you use a credit/debit card there.

https://www.irs.gov/payments

In the form you would indicate for "reason" that you want to pay "tax return", and then for "apply to" choose Form 1040/A/EZ, then choose the correct tax year.   At the end print out the confirmation for your records.

Or if you have an account (or establish one) at the Treasury's EFTPS system, you can make electronic payments there for tax due from a Form 1040, or for 1040-ES estimated tax payments throughout the year, etc.

https://www.eftps.gov/eftps/

I don't believe the New Filing Deadline is July 15, 2020.  The new deadline to pay taxes owed will be July 15.  The filing a tax return deadline will still be April 15, 2020 if you have taxes owed to avoid a failure to file penalty.

But since no one know for sure you should follow the IRS website for information - https://www.irs.gov/newsroom/whats-hot