Start by getting your online transcripts for the past 3 years, you want your tax return transcripts and your account transcripts. You need to see what you paid and what the IRS has on file for you.
https://www.irs.gov/individuals/get-transcript
As noted, the IRS figures penalties independently. Whatever Turbotax figured is an estimate, and is almost always wrong by at least a few dollars. And if you owe an additional penalty amount, the IRS will also charge interest. You need to look at your account transcripts to see if the penalties you paid were applied, and what additional penalties and interest the IRS calculated. You may correctly owe the additional amount.
You should have gotten letters or bills explaining this. They would have been sent to the address on your tax return. Did you move a lot? Or ignore the bills since you already thought you paid? Or never got them for some other reason?
You can apply for abatement of the penalty for good cause.
https://www.irs.gov/payments/penalty-relief-due-to-first-time-abate-or-other-administrative-waiver
I don't know if it would be granted in this case but maybe you could argue that you made a good faith effort to pay and didn't know there were additional amounts. Maybe you never got the extra letters and bills? Interest can't be waived by law, but if the penalties are adjusted, any interest will be recalculated.
