Szczesny has established himself as Arsene Wenger's first-choice goalkeeper in his first full season at the Emirates Stadium.
But the 22-year-old, who will make his 70th Gunners appearance in Saturday's Barclays Premier League game against Chelsea, is taking nothing for granted.
He told Arsenal Player: "I'm 22 years old and I don't think I'm an established number one. I still need to prove myself at this club.
"I do feel confident, but if we had this conversation in a few years' time and I've played 500 games for this club, I'll feel like I'm an established number one.
"I've played 65 games for the club, I think, and there's still a long way for me to go.
"So I don't feel like a proper number one - I still wear number 13 on my back!"
Szczesny claimed he had the "the best number two" in the country breathing down his neck in Lukasz Fabianski.
"I know I have to be at my very best in every game because he is waiting to take my spot," the Poland star said.
"It helps keep you sharp and to be at your very best."
He added: "I've never been short of confidence but when you actually start playing regularly and you feel like you're number one for the club, it gives you massive confidence knowing that the manager believes in you and trusts whatever you do.
"You know you can afford to make the odd mistake and you won't be dropped.
"It gives me a lot of confidence and that kind of shows in my performances.
"I believe I have improved, I've matured a lot in goal, I keep my focus for 90 minutes - which wasn't always the case in my early days.
"So the improvement is there but I'm hoping to improve with every single game and every single season at this club."
Despite his personal success this season, Szczesny admitted he could not be completely content.
"I would've been more satisfied if we had won any trophies," he said.
"My goal at the beginning of the season was to win trophies for this club and I haven't won any, so that's always a big disappointment.
"But it's my first full season in professional football, I'm enjoying it, and I think I've been quite consistent considering my age.
"Obviously, I could've done better on a few occasions but it's been okay."