Yes. Leaving "around" out in this sentence actually makes it sound more natural. "around" implies turning back in the opposite direction, for example, "We were going the wrong way, so we turned around."

Yes, you can say "I turned the corner." No, we do not say "I turned around the corner." Examples: I turned around. (I made a U-turn. I turned 180 degrees.) He turned the corner and found that he was going the wrong way down a one-way street, so he had to turn around quickly to avoid an accident. Turn right at that corner. I can't see around that corner. Can you? The post office is located around the corner. He lives just around the corner from me.

The answers above are right. I just wanted to point out that occasionally you might hear "round" used in place of "around". Both are correct, but "round" isn't used in this manner much in American English.