How to change the idea that supervsiors got about you?
The same way you would change anyone's perception of you for any other reason: Prove them wrong. 

If you started out lazy, then kick yourself into high gear: Get tasks done ahead of time; predict future work, and get it done before it's asked for. Turn out quality work at a high pace. This sounds hard because it is. Unfortunately, you may have burned good-will at this point, and you need to rebuild it. 

If you found things hard, or found that the supervisors expectations of velocity were unreasonable, discuss it with your supervisor. Discuss what background she may not know you have missed, discuss how you spent your time, and why things didn't get done fast enough.

Key here is the relationship. Your relationship sounds like it's on rocky ground, and if you're in it for another 3-5 years, then don't underestimate how much effort you're going to need to making sure that relationship doesn't break down.