Firstly don't try to lose weight too fast otherwise you will wreck your metabolism making future weight loss impossible. Losing 2 pounds (1kg) a week is as much as you should lose, so be happy with that.
It sounds like you eating is fine it's just your exercise we need to fix.
Strength training (weight) is very important because we need muscle in order to burn fat efficiently. Muscle is a very active tissue, so burns fat even while resting. Every pound of muscle you have will burn 50 extra calories per day. Don't worry about bulking up, that will not happen. You just want to add some lean muscle to your frame.
More importantly don't be obsessed with the scales, throw them away. Use the mirror, a tape measure and how you feel to judge weight loss.
If you burn more calories than you take in (achieved through a healthy eating plan), build up muscle through strength training and complete 30 minutes of solid cardio a day then weight loss is a certainty.
This following website gives you the tools you need to develop a healthy lifestyle and maintain it for a lifetime.
Good luck and all the best with your weight loss goals