All US citizens must use their US passports when entering or leaving the United States. This applies to both children and adults, as well as dual citizens holding foreign and US passports.
This is stated by the US Embassy in Sofia.
If you are leaving the United States with your Bulgarian or other foreign passport and cannot make an appointment with the embassy to apply for and receive a new US passport while in Bulgaria, you should plan for travel delays.
The Embassy will NOT issue emergency passports to US citizens who have left the country on their foreign passport solely as a matter of convenience, with the expectation that they can obtain an emergency travel document from the US Embassy to return to the United States. reported by the Embassy.