But the analogy never really worked.
Turkey, at least for now, has a more robust democratic system than Russia. Davutoglu's departure on Thursday from his post at the head of Turkey's government is a sign of clear differences between him and Erdogan over the direction of their country's politics. It's also possibly an indication of how Erdogan himself is growing impatient in his quest for more Putin-like powers.
Erdogan occupies what is supposed to be a nonpartisan, ceremonial role in a parliamentary democracy, where Davutoglu, a longstanding ally, was technically the leader. After serving as foreign minister for years while Erdogan was prime minister, Davutoglu became prime minister when Erdogan chose to run for the presidency in 2014. Davutoglu, it was imagined, would be the soft-spoken, bookish vizier to the tough, populist president.
Erdogan and his colleagues made no secret of their desire to rewrite the country's constitution — drafted by a military government in the 1980s — and expand the powers of an executive presidency. In late 2014, Erdogan unveiled his giant, new presidential palace in Ankara, a vast complex of 1,000 rooms that was meant to emulate the abodes of Ottoman sultans.
"Thinking big is not the work of dwarves," Erdogan said at the time, "which is not to offend dwarves. I love them too."
Despite internal unease within the ruling Justice and Development Party, known by the Turkish abbreviation AKP, and heated protests from political foes, Davutoglu and other party officials embraced their president's agenda, at least publicly.
Parliamentary elections last June threw a wrench in the works. Opposition to Erdogan's push for a presidential system, as well as the rise of a leftist, pro-Kurdish party, saw the AKP lose its parliamentary majority for the first time in over a decade. This prompted months of discord and acrimony as Turkey's parties proved unable to form a successful coalition government, an outcome which Erdogan's opponents claimed was precisely what the president wanted.
New elections in November followed a wave of instability and violence across the country. The polls restored the AKP's commanding majority in parliament and gave fresh momentum to Erdogan's plans to reshape the pillars of the Turkish state.
"The new Turkey will be built under the leadership of President Erdogan," Davutoglu declared from a balcony in Ankara on Nov. 1.
But his own role in this new Turkey is more difficult to fathom. Reports suggest the relationship between Erdogan and Davutoglu has been rocky for quite some time — with the latter frustrated by the former's clampdown on political freedoms, including the arrests of academics and journalists, and wary of scrapping the parliamentary system for a presidential one.
Davutoglu appeared to conduct recent negotiations with the European Union regarding Turkey's role in the region's refugee crisis without the full blessing of Erdogan, who was heavily critical of the Europeans in public statements. The clearest sign in the collapse of the tandem's relationship came last week when Erdogan loyalists voted to strip Davutoglu of the power to appoint provincial-level party officials, something that he, as the sitting head of the party, would be expected to do.
The prime minister departs at a time of great strain in Turkey. The country is struggling with conflicts on multiple fronts, including an escalation of a long-running Kurdish insurgency against the state as well as the chaos of the Syrian civil war, which has seen millions of refugees stream across the Turkish border.
Meanwhile, Erdogan, has become an increasingly polarizing figure, inveighing against the plots of enemies and would-be usurpers both abroad and at home. It's unclear who he will tap as Davutoglu's replacement.
"He will point to a more obedient and loyal personality," says Suat Kiniklioglu, a columnist and former Turkish lawmaker, in an email to WorldViews. "He was not fond of Davutoglu's foreign contacts and the potential of a successful conclusion of the refugee deal with the EU."
Kiniklioglu, who was once a member of parliament with the AKP, suggests some within the party will interpret Davutoglu's departure as evidence of Erdogan's inability to brook dissent, since Davutoglu "after all did not publicly diverge from Erdogan's discourse."
Davutoglu is hardly the first prominent AKP official to be sidelined by Erdogan. Others, including former president Abdullah Gul and former deputy prime minister Bulent Arinc, have long been whispered as potential figures to lead a palace revolt against Erdogan. But such is Erdogan's popularity and dominance over the AKP that it's hard to see any genuine fissures opening up within the center-right, religious nationalist party.
On his way out, Davutoglu himself stressed his loyalty both to Erdogan and the party.
"You will not hear one negative word from me about our president," he said.
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This post was updated.