Is there any way to pay your taxes online through turbo tax? Rather than mailing a check

No, online payments cannot be done through TurboTax

For federal taxes owed use this IRS payment website and Direct Pay - https://www.irs.gov/payments

not sure that's corrrect.  the info worksheet Part V allows you to specify a bank account for ach (direct debit)

however, your using online basic so I don't know if it has that feature.

if not the suggestion below is correct.  the states often have a similar method of paying so a check doen't have to be  mailed.   the iRS even allows the use of debit or credit card for a fee. 

If the question is can have a direct debit from your bank account to pay the taxes owed using TurboTax?  Then Yes for the Federal taxes owed and Maybe for the State taxes owed. 

Some states allow direct debit for state taxes owed when using tax preparation software, whereas other states do not.  If your state is one that does then the option for direct debit will be available for the state taxes owed.