As Aer Lingus is an Irish registered company. If they were to operate flights to Turkey they could only be from the Republic of Ireland. This is because Turkey has no aviation agreement with the EU.
In March the EU and Turkey did sign an agreement. However it wasn't full open skies
EUROPA - Press Releases - EU and Turkey initial civil aviation agreement
The European Union and the Turkish authorities have today initialled an aviation agreement which will remove nationality restrictions in the bilateral air services agreements between EU Member States and Turkey. This agreement will allow any EU airline to operate flights between any EU Member State and Turkey, where a bilateral agreement with Turkey exists and traffic rights are available.
Now I think UK-Turkey is open skies, at least on UK Regions-Turkey Regions routes. So Aer Lingus should be able to operate BFS-Turkey