Here are the reasons I think this is very unlikely:
1) Timelines for joining EU are very very long. Albania applied in 2009. Macedonia in 2004. Serbia in 2009. They are nowhere near close to joining, despite the "Western Balkans" being a priority in Brussels, both during the French and during the Bulgarian presidency of the Council of Europe. There is soooo many criteria to be fulfilled, particularly for poorer and more corrupt countries. Turkey is poor and corrupt.
2) Turkey will be an alternate center of power. The EU currently is like a solar system turning around Germany (or Germany and France, if you're being generous). Turkey, with it's 80 million+ citizens would immediately be the largest country in the EU Parliament. That will not be allowed to happen, no matter how many treaties and conventions they sign.
3) Turkey itself is not particularly interested in joining. And for good reason, in my opinion. They are starting to become a regional Power, and in the EU, they would be castrated.
4) For certain, while Erdogan is president, serious talks will not begin, so that will shave at least 5 more years from the timeline. So that would leave only 4 years until 2032. Not possible for this reason alone.
5) Any member of the EU can veto any candidate for any period of time. This happened with Macedonia twice - first Greece demanded that they change their name, and now Bulgaria demands they stop with the anti-bulgarian hysteria. In particular, various Dutch politicians have said that Turkey will never join the EU.
6) The culture is decidedly not European - turks are nationalists, a large fraction is Muslim, polygamy is still a thing, the military regularly does coups, large families are the norm, paying taxes is for suckers, there is a secessionist movement in the east.
7) There are 2 million Syrian and other refugees in Istanbul alone - how much do you think will go to Germany, Sweden, Denmark? Do you think the politicians there don't know that? They will invent whatever tortured reason against Turkey joining the EU, solely to prevent another huge wave of arab refugees. Not possible for this reason alone.
Even 5% might be too much.