Flights between countries are traditionally ruled by air service treaties between the countries concerned. To allow an Irish airline to operate between the UK and Turkey would require an amendment to the UK/Turkey deal and the Ireland/Turkey deal. The Turks, and possibly the UK, may want something in return from the Irish. Each side wants the maximum possible benefit for itself, and traditionally each country wanted to protect it's national airline
A good example is the Canada and the UAE. The UAE wants more frequencies but the Canadians won't agree to amend the treaty since it is unlikely any Canadian carriers will want them - indeed it could take away business from Air Canada, at least that is the fear.
In recent years we now have a common aviation area in most of Europe (and Morocco) and open skies. Some countries at least are changing philosophy from trying to protect their own interests to the 'free market' philosophy. Interestingly Canada has an open skies deal with Europe, but in this case Canadian airlines are better positioned to take advantage of this
Yes exactly, the basic concept is that where 2 countries do not have an open skies agreement, only aircraft registered in either of the 2 countries can operate a route between them. For example Ireland and Russia, only either Irish or Russian aircraft can operate between Ireland and Russia. The UK and Turkey are exactly the same, only either UK or Turkish registered aircraft can operate between the two countries. Therefore Ryanair, who are Irish registered cannot operate between the UK and Turkey. Similiarly Easyjet who are UK (or Swiss) registered cannot operate a route between Ireland and Turkey.