I found that the more time I spent riding, the more comfortable I felt navigating through traffic and making those turns you describe. What helped me the most was that I commute 3 days a week, so I got a lot of practice before getting into road biking.

As others have said, it's best to plan well ahead, look behind and move at the appropriate time when the coast is clear. One thing to realize is that your bike probably won't set the traffic signal sensor off, so if there isn't a car already in the left turning lane you may feel like a sitting duck out on that line. This happened to me recently where the left hand turn signal didn't change because I was the only object in the lane. Cars were whizzing by me at 50 mph so I just turned on the red because I felt it wasn't safe to stay where I was. In hindsight I should have just gone through the intersection and turned around.

It also helps to ride with an experienced rider to get a feel for how he/she navigates through the streets.

The more you ride the better you'll feel about it.