Final Fantasy VII
The Turks are effectively ShinRa's Black Ops department, as they are in charge of doing the majority of dirty underhand work. Officially, they are the Investigation Sector of the General Affairs Department.
In Final Fantasy VII, The Turks are lead by Tseng, with the assistance of Reno, Rude and their new recruit, Elena. While they aren't integral to the story, they help to drive it along by performing some horrible deeds that have been ordered by their superiors. The first of these deeds is the combined capture of Aerith Gainsborough and the destruction of Sector 7.
Tseng Is Badly Injured
The Turks' next job is to investigate the Temple of the Ancients, but it ends badly and Tseng is wounded. This leaves just Reno, Rude and Elena as the only known Turks in Final Fantasy VII.
Following the demise of ShinRa, Reno decides that they no longer care about the activities of Cloud and his group. They instead decide to follow their own orders and become their own bosses.
Final Fantasy VII: Advent Children
All four of the Turks that appeared in Final Fantasy VII also appear in Advent Children. However, due to ShinRa no longer existing, they technically aren't Turks anymore. Even so, they do still work for Rufus ShinRa and act as his bodyguards.
Their initial involvement sees Tseng, Elena and Reno extracting Jenova's remains from the Northern Crater. This doesn't end well though, with Tseng and Elena being kidnapped by Sephiroth's Remnants. Fortunately, they are rescued by Vincent and come to Rufus' aid when they are needed.
Now A Force For Good
Due to Rufus' new outlook on life, he uses the Turks to perform good. Although their primary role is to protect him, the Turks also fight against the Silver Haired Men and even manage to defeat both Loz and Yazoo despite being heavily out matched.
This was their last appearance in a Final Fantasy VII title though, as they did not feature in Dirge of Cerberus: Final Fantasy VII.
Before Crisis: Final Fantasy VII
One of the Turks' roles in Before Crisis is to recruit new candiates for SOLDIER; often by use of force or extreme "persuasion". The amount of Turks present in Before Crisis greatly exceeds that seen in the other games as ten playable characters were also included.
Before Crisis also sees multiple changes in leadership for the Turks, as Verdot, Heidegger and Tseng all take charge at various points. Verdot's reign is quite crucial to the story in the game as they fight against AVALANCHE, an eco-terrorist group.
Crisis Core: Final Fantasy VII
The Turks are employed in Crisis Core as more equal to SOLDIER and are often sent along to assist with their activities. This doesn't always work out well though, as some Turks are killed during an investigation into the activities of Genesis.
Despite orders to capture Zack towards the end of the game, both Cissnei and Tseng show that they don't blindly follow orders as they allow Zack to slip through their fingers. Towards the end, Tseng even orders the Turks to bring him in safely before the ShinRa army finds him, but unfortunately they are too late.