I think we'll see a semi-active June in the form of continued but gradually dampening protests, keeping the government busy for the next few weeks.
Then, two things happen: Ramadan starts and the seculars go on their summer holidays. Things should remain calm until well into October. That's when Turkey enters a period of heated debate and internal focus. The government will have to announce early elections.
The protests are serving as a safety valve for all the frustrations accumulated over the past decade. However, the protestors lack common political goals. For example, are they with or against the Kurdish peace? Being the master politician, Erdogan can easily work these differences ensuring that the "unhappy front" does not turn into a coherent political force.
There is no effective political opposition in the country. Maybe with the exception of the Kurds. It was interesting how a Kurdish MP played a leading role in starting the demonstrations but since then had to join the chorus for calling for calm. His imprisoned leader wants to ensure that 'while democratic rights are expressed, we should not let elements of the deep-state to take advantage of the situation.'
The protests are self-limiting in that sense.
Interestingly, the only dimension that can unify the protestors, and perhaps even many of the supporters of Erdogan is the level of Turkey's involvement in Syria. People want things to dial back a little. Reyhanli was a turning point in that sense.
I do think Erdogan is smart enough to realize that Turkey has no interest in going further than its main Western allies.
Internally, Erdogan may have lost the chance for a new constitution let alone changing the parliamentary system to a presidential one. He will focus on winning the presidency and before that ensuring that the position has extended powers.
Aside from that, the economy is sound. Nothing has fundamentally changed for the people who voted for Erdogan. If anything, some of his discontented supporters will now be more inclined to think Erdogan deserves continued support.
For Erdogan's friends and foes, the fundamental reality is he has no alternative. Everybody will have to work with that for now.