I just noticed that while running, I feel like my whole weight is in my legs, which is very uncomfortable, and my legs are stirring on the sides. Is that normal? Does it just mean that I am supposed to lose weight (1m70/ 65kg) or is there another reason?
I think heavy legs are largely a factor of not enough circulation to you legs, making them weak & thus feeling heavy.
Try longer warm ups and stretches, (give it about 15 mins) before and after each run to help strengthen your legs (especially since you're just starting out), drink lots of water (during the day, 30-40 mins before you run and after) & potassium helps circulation as well.
Let me know if you want any suggestions on some good stretches. When you warm up, stretch and hold the position for say 10 counts & don't bounce. When you hold the stretch, it allows the body to have time to release lactic acid and etc.
I think I've been through ever leg pain possible, knees, shins, quads, & feet. lol
Also how are your shoes? Improper & heavy shoes will definitely feel like weights on your legs.
Try a cool down after your runs, and a really good 10-15 min session of stretching/yoga or even a steam room+stretching and see if that helps your legs from feeling like lead the next day.
Its just feeling like that due to the lactic build-up which is why your legs feel better after you start running again, because it gets the blood circulating and moving the lactic acid out, which is what a warm up and stretching will help do as well
Hope this helps :)