The main two topics on Tuesday's agenda were Trump's decision to send weapons to Syrian Kurdish forces of YPG and Turkey's request for the extradition of Fethullah Gulen, a US-based Muslim leader.
Gulen, who has lived in Pennsylvania since 1999, is accused by the Turkish government of having planned the failed military coup in 2016.
The US announced on May 10 that it would supply weapons and military equipment to YPG units of the SDF alliance battling the Islamic State of Iraq and the Levant (ISIL, also known as ISIS) group in Syria, despite strong opposition from Turkey, a NATO ally.
The Kurdish elements of the SDF, which is backed by the US-led coalition, are from the YPG, or the Kurdish People's Protection Units, the main faction battling ISIL.
"We will support Turkey in the fight against ISIS and the PKK and we will ensure that they have no safe quarter in the region," Trump said at Tuesday's joint news conference with Erdogan.
For his part, Erdogan said: "Taking the YPG and [its political wing] PYD into consideration in the region will never be accepted and will go against a global agreement that we have reached.
"We should never allow those groups to manipulate the religious and ethnic structure of the region making terrorism as a pretext or excuse."
Turkey says the YPG is linked to the Kurdistan Workers' Party (PKK) inside Turkey, which has waged an armed campaign since 1984 that has killed more than 40,000 people.
To supply weapons to Kurdish units in Syria is "unacceptable", Turkey's deputy prime minister said in a TV interview.
The Pentagon says the YPG is "the only force on the ground that can successfully seize Raqqa in the near future".
For its part, Turkey is demanding that the Raqqa offensive is carried out with the Free Syrian Army and not the SDF.
International news media quoted Erdogan as saying before boarding the flight to Washington, DC: "We are either strategic allies and decide together, or we are not, and then we have to go our separate ways."
Al Jazeera's Patty Culhane, reporting from Washington, DC, said Erdogan insisted that he reserves the right to fight "terrorism" in the region on all fronts.
"This meeting will have a direct impact on the conflict in Syria and the Syrian people," she said.
Marwan Bishara, Al Jazeera's senior political analyst, said: "So far, President Trump doesn't really have a clear foreign policy in the Middle East, so Erdogan wishes clarity, engagement and cooperation.
"There's a lot of discord, but there is also a chance that the two like-minded people […] will work together. They can both benefit from each other.
"Erdogan would love to get Trump and Putin to work together with Turkey and thus eliminate Iran from the equation."
As for Gulen, Turkish officials have officially requested his extradition from the US to stand trial in Turkey.
Erdogan had sent two top officials to Washington, DC, before his own visit to the White House.
Hulusi Akar and Hakan Fida, respectively the Turkish military and intelligence chiefs, held talks with senior US officials.
Turkey's Anadolu news agency said more than 100 supporters welcomed Erdogan as he arrived at Trump's guest home in Washington, DC.
As he pulled into Blair House, supporters chanted Erdogan's name and "red and white", the colours of the Turkish flag.
Following Erdogan's arrival, Turkey's ambassador to the US thanked supporters, some of whom he said travelled to Washington, DC, from as far away as California on the West Coast.
"This unity and solidarity, especially in these days is very important," Serdar Kilic said.
A meeting with Turkish-American business and religious leaders at the Turkish ambassador’s residence in northwest Washington, DC, was scheduled.
Gunay Ovunc, a second-generation Turkish-American who co-chairs the Turkish-American National Steering Committee, said: "Here today, we are beyond party lines.
"We have transcended those divisions."
Source: Al Jazeera and news agencies