Hi OP, it's great that you have looked into this and want to do something about it. There's so many big kids about and I think many don't realise or bury their heads about it.
My child was quite overweight, but now they are 'normal' according to their BMI. I couldn't really see what I was doing wrong but upon reading around processed foods and the Mediterranean diet I could see there was room for improvement.
The major changes I made were to remove as much highly processed food as possible and introduce more whole foods in their diet.
Cereals were cut out for a long time and now are just an occasional treat. Cordial/squash and fruit juices were cut out and water and milk became the default. Yoghurt offered became Greek yoghurt with berries mixed instead of sugary yoghurts. White carbs were replaced with wholegrain options. I even started making my own bread. Vegetable intake was increased so that pretty much every snack and meal included it. For meals I'd try to fill half their plate with veg. Processed meats like bacon and sausage were cut out. Fish fingers and waffles cut out. Instead I give just normal meat and fish, potatoes, wholegrain pasta, brown rice or couscous etc. I've introduced more lentils, beans, chickpeas and nuts. I don't give a pile of spuds with every meal. I try to treat potatoes like any other veg and give a smaller portion, and increase the amount of less starchy veg.
Once they get used to real foods they will be fuller for longer and not overeat processed stuff that's designed to be overeaten. They won't have blood sugar spikes and crashes without sugar and white carbs.
Snacks are fruit, nuts, crackers, toast, veg sticks, hummus, cheese, breadsticks. I find the least processed crackers and breadsticks I can find. There is a fruit bowl always topped up with apples, bananas, oranges and whatever else looks nice that week.
Breakfasts are more varied and not just a white carb and some fruit. I give homemade wholegrain bread with various healthy toppings, porridge, omelette, even homemade baked beans. Homemade granola with berries and Greek yoghurt goes down a storm.
If we go out I try to bring a packed lunch instead of relying on crap food devoid of nutrients that's sold as kids food in most venues.
Basically anything too processed is now an occasional treat whether that's shop bought croissants, processed meat, takeaway food or a bar of chocolate, and it seems to be doing the trick. My child has grown in to their weight and now has a normal BMI and looks so different to older photos of them where they were obviously overweight.
I would say don't be afraid of talking to your child about the nutrition value of food and how we should try to eat a healthy and varied diet. They cover this sort of thing in school so it shouldn't be a foreign concept to them. If you have any cookery books in the house you can get them out and look at the pictures, you might be surprised to find what she thinks looks good and would be prepared to try. If she doesn't already help you in the kitchen I would definitely suggest this, in my experience it helps kids become more familiar with new ingredients and more adventurous in what they are prepared to try a bit of. Best of luck to you both!