Chancellor Angela Merkel said Germany needed more evidence if it was to class the movement of US-based cleric Fetullah Gulen as terrorist organisation.
Speaking at a Berlin news conference alongside Turkish President Tayyip Erdogan, she said that Germany needed more material if they were to proscribe the group that Ankara blames for a failed coup attempt in 2016.
"We take very seriously the evidence Turkey provided but we need more material if we are to classify it in the same way we have classified the (Kurdish) PKK," she told reporters.
The issue is one of the key areas of contention between the two countries as Mr Erdogan began his three-day visit on Thursday. During the press conference, Mr Erdogan called on Berlin to extradite what he said were the hundreds of supporters of the cleric living in Germany.
He also said that Turkey had the right to request the extradition from Germany of Can Dundar, former editor of opposition newspaper Cumhuriyet, describing him as an agent who had been convicted of espionage.
Mr Dundar and a colleague were sentenced in 2016 to five years in prison for publishing a video purporting to show Turkey's intelligence agency trucking weapons into Syria. They were released pending appeal and Dundar left the country.
The two leaders also agreed to aim for a four-way meeting with themselves and the presidents of France and Russia in October to discuss the situation of the rebel-held Syrian region of Idlib.
In an editorial in the Frankfurter Allgemeine daily, Mr Erdogan said he wanted to "turn the page" on a long period of tensions, sparked by Berlin's criticism of his crackdown on opponents after the failed coup, highlighting the economic importance of his trip to Turkey.
Mr Erdogan is eager to improve ties with Europe's biggest economy and secure investments to shore up a struggling economy at home. The meeting came a day after Germany beat Turkey to become the Euro 2024 host nation, following a tight race that took on political significance when Mr Erdogan fanned accusations of German discrimination in football.
His state visit to Germany, complete with military honours, is Erdogan's first there since becoming president in 2014 and comes as he is sparring with US President Donald Trump and the Turkish economy is in rapid decline.
But critics, including rights campaigners and German politicians, are angered by the red carpet treatment for a leader who has built an increasingly authoritarian reputation and just 18 months ago accused Berlin of "Nazi practices".
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Ms Merkel herself has repeatedly stressed the importance of good relations with Ankara, a partner she relies on to help stem the flow of migrants to Europe.
But the hostility towards the visit comes at an awkward time for the veteran chancellor, who can ill afford any missteps after being weakened by a slew of crises that have rocked her fragile coalition.
Europe's de facto leader last week was forced to backtrack on a decision to promote a domestic spy chief who was under fire for his alleged far-right links, prompting Ms Merkel to admit she had misread the public mood.
Mr Erdogan critics have vowed to take to the streets across Germany to protest everything from Turkey's record on human rights and press freedom to its offensive against Kurdish militia in Syria.
Demonstrators are also planning to protest in Cologne on Saturday where President Erdogan will open one of Europe's largest mosques, commissioned by the Turkish-controlled Ditib organisation.
"Erdogan wants a fresh start with Germany. This is an opportunity," the Sueddeutsche Zeitung said, urging Ms Merkel to push Ankara to end its repressive tactics and free the five remaining German-Turkish nationals considered political prisoners by Berlin.
"But we can't just forget everything that happened. It could take years to rebuild trust," it added.
Relations between the two NATO countries plummeted after Turkish authorities arrested tens of thousands of people in a mass purge over the attempted putsch against Mr Erdogan.
But a gradual rapprochement began after German-Turkish journalist Deniz Yucel was freed earlier this year. He still faces terror-related charges in Turkey however.
Germany is home to a three-million strong Turkish community and observers said Ms Merkel now faced the delicate balancing act of accepting Mr Erdogan's outstretched hand - without glossing over their disagreements.
President Erdogan for his part said he would use his trip to urge Germany to show "the necessary support" in fighting the fight against "terrorist groups" like the Kurdistan Workers' Party (PKK) and the movement of Muslim cleric Fethullah Gulen, whom Ankara blames for the coup.
In terms of economic cooperation, Der Spiegel weekly reported that German conglomerate Siemens was in talks to lead a potentially $40-billion deal to modernise Turkey's rail infrastructure.
In a sign of the contentious nature of the visit, several opposition politicians have vowed to boycott Friday's state dinner in Mr Erdogan's honour. Chancellor Merkel too will be absent, although her office insists it's not out of the ordinary for her to skip such events.
Ms Merkel and Mr Erdogan are scheduled to hold a second round of talks on Saturday.
