It seems you are used to brakes that are not so effective, and when riding with disc brakes you pull them on just as hard. Most people have learnt how effective disc brakes are and don't pull them on so hard, so don't have the same problem. You will probably get in that habit in time.

Anyone who has cycled a lot could probably tell you stories. I had a situation many years ago while riding to school. As I went around a corner near school, occasionally there would be another student crossing the road. I would just swing around behind them. One day, one student stepped back. I was too close to swerve any further, so I locked up the brakes and went over the handlebars. I did not hit the student.

It is also smart to know which brake to use. I use the front brake when going in a straight line. If when going in a straight line, I need to stop suddenly, or am going down hill, I use both brakes. If it is down a steep dirt track, or on any surface where you don't get a lot of traction, you cant stop suddenly, you need to keep your speed down, and only slow down gradually. I use only the back brake while turning. If it is not a sharp bend, I use both. Ensure you don't brake too hard on corners.