No they are not the same thing they are not even spelled the same. Do not enter the conversion as a contribution.
A conversion is a type of rollover. You need to report it as a conversion. The taxes you paid were only an estimate. You also report the taxes you paid and they go to your credit. Your final tax bill is computed based on all your income, deductions and credits, and if you overpaid, you get a refund; if you underpaid, you owe an additional payment.
You should get a 1099-R from the first IRA. When you enter it in Turbotax, the program will ask what you did with the money, and you will check the box for a Roth conversion. If you leave the conversion off your tax return (because you think you already paid the tax) you will get a nasty letter from the IRS.
Conversions are separate from contributions. You can contribute up to $6000 (depending on your income and other factors) and your contribution limit is not affected by any rollovers or conversions you might have made.