Turn has many meanings. One meaning is to rotate or spin around a central point. Another is to change one's direction of travel. Turn can also be used as a noun to describe that change in direction.
Ex: The earth turns on its axis, completing one revolution every twenty-four hours. At the same time, it rotates around the sun, which is in turn rotating around the centre of the galaxy.
Ex: After you pass the baseball stadium, you'll have to turn left at the traffic light, and then make your first right turn after that. Our house is down at the end of the road. You can't miss it!
One common expression using turn is turn up, which means appear or arrive. Another is take a turn for the worse, which means that things started going badly.
Ex: I was getting ready to go out last night, and guess who turned up at my door? Tommy G, from the old neighbourhood. I had to stay and talk to him for a while, even though my boyfriend was waiting for me, you know?
Ex: Everything was going well on our road trip. We were all getting along, we were making good time, and the car was running well. Things, however, took a turn for the worse very rapidly this morning when black smoke started billowing out from under the hood.
A curve is a gentle and gradual bend, such as the way a road's direction changes. At its simplest, it is a line that is not straight and has no angles. It can also be used as a verb, such a line is curved.
Ex: There are a series of S-curves in the road that Howie lives on, and they can be treacherous in wet weather. It is wise to slow down considerably when approaching them.
Ex: Mary Jean suffers from scoliosis, which means that her spine, which is relatively straight in most people, is significantly curved. This can present some serious health risks if left untreated.
In baseball, it is possible to throw a pitch in such a way that the ball's path curves through the air. Such a pitch is called a curveball. Metaphorically, if you are confused by any sudden unexpected event or trickery, you have been thrown a curve.
Ex: Roger was the best pitcher on his team. His fastball was faster than anyone else's in the division, and his curveball was a thing of beauty—no one could hit it!
Ex: Martin and Sam were having a great time reminiscing when Martin reminded Sam that he had borrowed one hundred dollars from him in 1992, and had never paid it back. He really threw Sam a curve. So Sam, caught by surprise, paid him then and there.