Actually, I think your trainer is being responsible. Look, for every extra pounds of body weight, you add an extra 4 pounds of pressure on your knees when you walk---when you run, it's an extra 10 pounds. So if you are only 25 pounds overweight, but you run, you're adding an extra 250 pounds of pressure on your knees.. and that's on top of what would be your normal weight.
The people above who are running now, will be paying for it in 20-30 years, with increased risk of osteoarthritis and knee replacement surgery.
There are plenty of low-impact cardio activities you can do if you really want to work on that. Try a spinning class or the elliptical trainer.
Knee Health - Bad Habits That Could Be Destroying Your Knee Joints - RealAge
You might also check into burst training...long bouts of cardio is on the way out as a health fad, with more and more studies showing it's not really effective, and does some long term damage.
Here's a link. There are others on the side. Dr. Al Sears also has a book out on this.