The Germany international midfielder completed a deadline day move to the Gunners from Real in a deal believed to be worth £43m - smashing the Arsenal club record for a new signing.
The 24-year-old admitted he feared dropping down the pecking order at Real after the signing of Bale, which he viewed as a lack of faith in his abilities from coach Carlo Ancelotti.
But Ozil believes he can thrive in the Premier League under Arsene Wenger, and he is looking forward to the challenge.
In an interview with the German Football Association's official website, Ozil said: "At the weekend, I was certain that I would stay at Real Madrid.
"But afterwards I realised that I did not have the faith from the coach or the bosses. I am a player who needs this faith and that is what I have felt from Arsenal, which is why I have joined.
"I am really looking forward to it because I have the faith of the coach. I had spoken to him at length on the telephone, he explained his plans and that he has faith in me - that is what I need as a player.
"I am looking forward to the new challenge there (Arsenal). I have already heard that they have super fans, the city is great and the team is fantastic.
"Wenger gives me the faith and I can develop further. I know what I can do and I know that I could make the grade with any club in the world because I am so convinced in myself.
"But if I don't feel people have faith in me, then that is why I had to leave and at Arsenal, I feel I have this fully. I am sure the new coach is going to develop me further and I am also really looking forward to playing in the Premier League."
England and Arsenal striker Theo Walcott was excited by the prospect of playing alongside, rather than against, Ozil.
"I played against him for the U21s when we lost 4-0 and he showed his qualities then and he has shown them for many years for club and country," Walcott said.
"He is a great addition and I am personally looking forward to playing with him."