The worksheet calculations in TurboTax use the first and last balances to figure the average balance, whether it's 2 months or 10 months doesn't matter. Using the balances from your mortgage statements each month and figuring the average may result in a different figure if for example you had months of no principle payments or months of large principle pre-payments between the first and last. However, if the principle payments are steady both methods will end with very close amounts.
As Publication 936 shows you can use other methods of figuring your average balance on a loan. In the "forms mode" on the "Ded Home Mort" worksheet you can override the Average Balance entries with your own figures. Be advised overriding TurboTax calculations will nullify the accuracy guarantee offered by TurboTax.
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