Yes, Turkey is free to project power now. But Erdogan's Islamist government makes that power projection a problem more than an asset for the West.
Turkey’s successful military intervention to preserve Libya’s Government of National Accord (GNA) marks a turning point for the security architecture of the Middle East and North Africa. Turkey’s new capability to project military power far beyond its coastal borders—a paradigm shift enabled by the rise of its defense industry—has made Turkey’s strategic orientation one of the most significant determinants of the region’s geopolitics. How Turkey calibrates the congruence between its hard power instruments and its strategic orientation now constitutes a factor of the utmost consequence for the strategic calculus of the entire Mediterranean basin.
Yes, the eastern Mediterranean Sea is now a point of competition.
The reason Turkey can do this is the complete collapse of the Soviet ground threat to Turkey:
Without the Soviet ground threat looming over Turkey from Bulgaria, Georgia, and Armenia, the Turks can focus on their Ottoman restoration strategy and push to dominate the eastern Mediterranean Sea.
And Turkey is making plays for influence east (and south) of Suez.
I guess the scale of Turkish militarization of their ambitions and how much they step on former allies' toes will determine whether NATO ejects Turkey from that defensive alliance. Or whether Turkey withdraws on its own.
And then it gets interesting. With Turkey in opposition to NATO, Russia, Israel, the bulk of the Arab world, and Iran, too, Turkey will need to align itself with China just for the nuclear umbrella until Turkey can get their own.
Israel's new "Saar 6" corvettes will improve Israel's ability to protect their offshore gas platforms
offers another complication to Turkish ambitions. Supported by Israel's air power, Israel is a factor. The article mentions Hezbollah threats but with Turkey's hostility to
Israel and new offshore ambitions, Turkey is a much bigger threat to
Israel in scale if not in day-to-day security concerns.
Do read all of the initial article which has quite a lot of information.