The capital club have won trophies in each of the last four seasons, winning two UEFA Europa League titles, one Copa del Rey and two UEFA Super Cup crowns.
And Diego Simeone's men are in contention for yet more honours following a fine first half of the campaign.
Atletico are level at the top of La Liga with defending champions Barcelona and are still very much in the running to retain the Copa del Rey and face a Champions League knockout clash with stuttering Serie A side Milan.
Valencia are their opponents in the second leg of the last 16 of the Copa del Rey on Tuesday, a tie which is finely poised at 1-1 after the first leg, and Turan believes all three trophies can be secured.
"It is clear that we have to fight until the end for the three titles that we have ahead: the Cup, the League and the Champions League," Turan told AS.
"This Atleti is (as good as) a Bayern (Munich), Real Madrid or Barca. Being a soccer player of Atletico Madrid is cause for pride and happiness."
Still, the Turkey international is keeping his feet on the ground and taking every game as it comes, a mantra often repeated by Simeone.
"At the moment, the most important thing is to beat Valencia. And in a few days I will say the same for our Liga game against Sevilla," he added.
"We are thinking only what we have ahead. For us every match is a final, no matter what competition it is. We are not arrogant, we are rather humble."
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