Don't believe what Trump said. Turks and Kurds are not fighting for 200 years or something. Turks and Kurds are not fighting, those ethnic based conflict story makers are just fantasizing.
Turkey and Ottoman Empire had great natşonal and religion-based enemies but just like what I said about Greece, It all became history. After War of Independence, after pushing Greeks back to Aegean Sea, It was over for us. After that there were tensions between Turkey and Greece though, solely based on Aegean Islands and Cyprus.
And there were Armenians, no not the genocide, even before Kurds, they wanted to take a slice of old Ottoman Empire's east and establish their country. They literally hit Ottoman on the back at WW1 with revolting, mass raiding eastern Village killing innocent people etc. They were supported by Russian Csar and British. Ottoman state made their own take on them, Turkey did some other. But shortly after that ASALA founded. It sometimes referred as PKK's predecessor. After PKK's open insurgency, it is found that old ASALA cells were influenced PKK, helped to organise and even literally join the fight againts Turkey.
Kurdish people were never an organised population anyway. They were living under regional tribe chiefs. Some of them still exist today. Tribe chief is a patriarch and his rule is definitive. When there is an election, political parties goes and convince him only and he gives orders to his tribe to vote which party he decided to vote. Same goes for HDP and even PKK. Some tribe chiefs were sided with Turkish state, which made them targets for PKK assaults. There were a lot villages raided like that.
Turkey initiated the mostly volunteer based village guard system because of that. https://en.wikipedia.org/wiki/Village_guard_system
So back to the main topic, no, Turks and Kurds are not fighting. Legitimate state is fighting againts illegitimate factions and terrorist groups. There were this Kurdish rebellion even in early life of modern Turkey but they wanted to re-build İslamic Caliphate because Turkey decided to be secular. https://en.wikipedia.org/wiki/Sheikh_Said_rebellion
Obviously, as a wise man, Atatürk saw the potential integrity issues with Kurds being a ethnic minority under Turkish government. But he didn't decide to wipe out or mass murder anyone. He just tought it is an open invitation for western powers to rally and help Kurds and Armenians build their own state 'inside' the borders of Turkey. So yeah everyone wants best interest of their country type of thing but Ottomans were fighting for years, especially in WW1 it took a great toll on the people. Turks saw what a monster is the other imperialist powers.
You can see how harmful Sykes-Picot Aggreement on middle east today and Treaty of Sevres was just it's even worse Anatolian share. https://en.wikipedia.org/wiki/Treaty_of_Sèvres
Lucky for us, we had Mustafa Kemal and his friends. He bring the broken, ill man of Europe to a modern, secular republic of Turkey. His idea of Turkish borders were based on Misak-ı Milli aka National Oath. https://en.wikipedia.org/wiki/Misak-ı_Millî
At that time, Kurds were not even at a place to form large mass of population and they were citizens of Ottomans Empire. Most Kurds also fought in WW1 or other wars with Turks and many other ethnic groups, gladly died protecting the country.
But I have to also note that especially after Atatürk'ü death, people who rule the state and sometimes government took Atatürk''s concerns to an another pedestal. Talking Kurdish in public got banned for instance. Most of them were uneducated, most of them didn' t even know any other language but those people who were making decisions thought that this was what Atatürk would do and letting them speaking Kurdish is bad for unitery Turkish rule. This, unfortunately led Kurds to hating Turkish state.
But again, I have to note that this topic is much more complicated then this. Those wrongdoers were alleged with Gladio and Anti-Soviet people. Socialism was on the rise with Turkey's population, a NATO ally. Someone hit Kurds because of that but we also have to see that someone also hit socialist Turks too. Turkish history of politics is filled with controversy especially after 1950s.
I encourage you guys to read about it. 1960 coup, Turkey-Greek relations, Cyprus, Johnson letter, Gladio, Counter-Guerilla, U.S. backing Islamists againts socialist etc. These all vital element to understand both Turkey and the region.
PS: Sent from mobile, sorry for spelling mistakes.
Ah also kinda nice and informative article on the topic: https://www.vox.com/world/2019/10/16/20908262/turkey-syria-kurds-trump-invasion-questions