Turian status:
1)Enemies - Decades ago
2)Enemies with restraint - Decades ago until ME1
3)Friends with restraint - ME1 to ME3
4)Friends - ME3+?
Each being the first ones on the Council to want to join forces and offer aid to the other during the Reaper war, is kind of like watching two brothers who don't like each other, but are still willing to get into a fight to back the other one up.
Exactly. Even in the micro scale, Garrus' initial story was of a Turian who didn't exactly want to fit the rules/status quo, and so joined a Human against another Turian.
Between the civilizations, there was initial open hostility, then more closed hostility (the older Humans know this, and its shown through characters like Pressley), then grudging respect as time goes on and as the Citadel was saved (varied in depiction if Council was saved), growing into a stronger respect as and after the Reaper hit and are defeated.
The laughed-at supposed leak rumor of 'Mass Shift' described a remnant Alliance becoming a merging of Human and Turian militaries and a surviving Garrus put at the top or near top of its ranks. Out of all its content, this idea was one of the most believable ones. The plot has Humanity forming close ties with them no matter what, even if they may 'bicker like siblings'.
Of course there's other species you can form legendary relations with, at least depending on choices. Arguably you could include all of these: Krogan, Quarian, Geth, Salarians and Asari. But for MOST ties to HUMANITY, I'd say Turians. The Tuchanka and Rannoch stuff had a distinctly 'ties with Shepard, not necessarily all Humans' feeling to me.
Second place might be Asari. The Asari garner influence just be hanging around a space and being themselves, after all.
EDIT: That said, I get the growing sense in the trilogy that Humanity is set to expand ties with several other major species, or really all or almost all of the ones Shepard deals with - barring future plot and its twists. Quarian and Geth and Krogan (and Salarian) involvement in taking back Earth and saving the Galaxy isn't to be taken lightly. There is both/either the thing of a more united galaxy with each other, and a galaxy where Humanity is arguably at or near the 'top' (except again, in a cooperative united galaxy, at least earlier post-war). So we're getting closer ties all over the place, even if its arguably with Turians the most.