Bush said: "The Turks, the Americans, and the Iraqis including the Iraqi Kurds share a common enemy in the PKK.
"It is in nobody's interest that there be safe haven for people who have the willingness to kill innocent people."
Turkey has so far resisted calls from Iraq, the United States and other nations to say when it would end its week-long operation.
Gates had earlier held talks in Ankara, Turkey's capital, with Recep Tayyip Erdogan, the country's president and Abdullah Gul, Turkey's president.
He also met with Vecdi Gonul, his Turkish counterpart, and General Yasar Buyukanit, the head of the country's military general staff.
Gates said: "In the sessions that we had, there was no specific mention of a date. I think they got our message through."
Speaking in India on Wedesday, Gates had told journalists: "I measure quick in terms of days, a week or two, something like that. Not months."
However, Buyukanit said: "A short time is a relative concept, it could be one day or one year."
Gates told Turkish leaders that providing more information about the numbers of troops involved, and the limited area in which the operation was taking place "would help correct a lot of misimpressions and potential misunderstandings."
He said: "Based on what they told me it sounds like the size of the operation is probably proportionate.
"They described the operation, and the president and prime minister in particular, described some of the initiatives they have taken in a non-military arena.
"And their concern is finding the balance in terms of how much they can say publicly without compromising their military operations, particularly in the context of timing.
"They've clearly reached out to the Iraqis, they just had a special mission return late last night from Baghdad, so I think they are beginning that part of the process."
But Barham Saleh, Iraq's deputy prime minister, warned Turkey of any prolonged incursion and demanded the attacks end.
Both nations have been close allies and have shared intelligence reports on operations against the PKK.
Gates said: "I think that those interests are probably not advanced by making threats, by threatening to cut off intelligence."
Turkey's military has said it has killed 230 PKK fighters since the offensive began
The PKK disputes that number and says it has killed 81 Turkish troops during the fighting.
Cemil Cicek, Turkey's deputy prime minister, played down mounting international concern about the cross-border offensive.
Cicek said: "No one should be concerned. We will leave Iraq as soon as we are done. We are not intending to remain any longer than necessary in minus 26 degree temperatures."