Being "responsible for" the expenses of your dependents does not mean that you can file as head of household. The requirements for head of household filing status are somewhat complicated, as you can see from what Bsch4477 posted. If you answered the questions in TurboTax accurately, TurboTax is most likely correct in saying that you have to file as single. If you want to check, you can use What is my filing status? on the IRS web site, or see "Head of Household" on page 8 of IRS Publication 501.
