The 28-year-old moved to the Gunners from the New England Revolution in a deal worth up to $10 million and started the club's first preseason friendly on their U.S. tour in Saturday's 2-0 win over Everton in Baltimore.
Aaron Ramsdale has established himself as Arsenal's first-choice goalkeeper under Mikel Arteta and Turner appears to face a difficult task in dislodging him, despite being named MLS Goalkeeper of the Year in 2021.
Turner is currently vying with Zack Steffen for the No. 1 jersey in Gregg Berhalter's side with the World Cup just four months away.
"Opportunities like this for players from the U.S. are few and far between," Turner said. "I wouldn't say I'm worried or it's a gamble. This is the obvious progression in my career.
"Getting my foot in the door overseas is a lot easier said than done. I've been playing well in MLS for the last three seasons and this was the first concrete offer I had.
"Playing consistently in MLS did not guarantee me the starting XI for the U.S. national team. I needed to shake it up to take my game to the next level and playing in the Premier League with these guys has already shown me some massive improvements.
"You have to think about your entire career. You can't make decisions based on one World Cup. Injuries happen and athletic careers are finite. I've wanted to take this game as far as I can, so to be able to represent a club like Arsenal is a dream come true."
Asked how Arteta had described his possible role at the club before signing, Turner continued: "Mikel told me I am here to challenge for the starting role. We're not in this business to just accept being second.
"We all want to battle for time on the pitch. So the mentality he wants for me is to push Aaron, to push myself and to push the guys in the locker room. There's a lot of young guys bopping around and I am a senior player here at 28. So I can be someone those guys can rely on for advice on and off the pitch.
"Every step I have gone through in my career has been a big step. Going from high school to Uni, then Uni to MLS, then MLS to the national team and now to the Premier League. The players I am with every day are very talented. They bring a consistency, an intensity and a different sort of intelligence in the way they play the game.
"The other big thing is the standards they hold you to every single day. The coaching staff and the players.
"Sometimes the training regimen can be a bit more casual in the U.S. Maybe what I was used to at the Revolution was a little more casual. That has been a bit of an adjustment, but I was eager for a new challenge and this is definitely what I have."