Here's the problem: If you are audited (most people are never audited), it is up to you to prove your prior contributions, and that you are only withdrawing your prior contribution basis. You can make a reasonable guess on your return, so that Turbotax won't tax you now, but if audited, the IRS wants more than guesses.
Contributions are reported to the IRS on form 5498. You should be able to get transcripts of the last 7 years of form 5498 from your account online.
https://www.irs.gov/individuals/get-transcript
Going back farther, if you did not save copies of your form 5498 and did not save quarterly or annual statements, you can try contacting the prior brokers. Or see how far back your checking account records go that you can document deposits and transfers. You will have to do the best you can.
