Turkish visa application states applicant must have round-trip ticket with Turkish Airlines, Pegasus Airlines, Onur Air or Atlasglobal Airlines. Does British airways flight count?
Citizens of certain countries can apply for a 'conditional e-visa' or evisa on arrival which carry certain conditions.
If you're a citizen of one of those countries and are applying for such a visa you must use an airline that has established the correct protocols with the Turkish Ministry of Foreign Affairs. For now, those are only the four airlines you've listed in your question. Flying with British Airways doesn't count.
You can find details on this Wikipedia page
Be sure you're applying for the right visa.