"when the taxable amount on the conversion is not due until April,"
Nope, sorry. All income taxes are pay-as-you-go. The IRS assumes that your income is spread out over the entire year and expects your payments (withholding or estimate payments) will also be evenly spread out through the year. If you made a conversion in January, you would owe a single estimated tax payment on April 15, or you could divide it into 4 equal payments to be made before April 15, June 15, Sept, 15 and January 15. Failing to make those payments can subject you to an underpayment penalty even if you pay in full when you file.
To minimize the amount of penalty and interest, you can make an estimated payment now at www.irs.gov/payments. That will result in less compounding of the interest and penalty.
Also, if Turbotax allows, you can decline the option to include a pay a penalty, and wait for the IRS to send you a notice and a bill. At that point, you can request an abatement for cause, or as a first-time forgiveness if you never owed a penalty before. It's harder to ask for the waiver if you already included a payment with your tax return.
