Made an early Roth IRA distribution. Can I estimate prior year Roth contributions made over 10 years ago since I no longer have records? If so, how?

[afairless]

I funded my IRA starting in 2000 through 2012 and I cannot locate any bank statements or tax records. My financial institution so longer has records prior to 2018 and the IRS no longer has records prior to 2015. Can I estimate my prior year contributions to my Roth so that I don't have to pay taxes on the full amount of the early distribution? What are the acceptable methodologies in order to estimate such Roth IRA contributions?

[Opus 17]

There is no guidance on this I can give you.  If you are audited, you are not entitled to any tax break you can't prove with adequate, reliable records, and the auditor has a certain amount of discretion in what to accept, but they are going to want to see something.  Of course, most taxpayers are not audited. 

[fanfare]

If your distribution is much less than your contributions, enter any amount,

Enter the amount of the distribution as contributions and TutboTax will set your contributions as exhausted.

This will remind you to get a more accurate number if you plan to take regularly from your Roth IRA.

@afairless