Diane, You certainly had success growing those from seed! Yes, nitrogen does stimulate green growth, but many lawn fertilizers are especially high in nitrogen and/or formulated specifically for green, lush grass. So if it washed into the bed on the right it would have the same effect on those cosmos. I'm also looking at the soil there and wondering if it is better (richer, loamier) than the strip along the fence, which might have previously been grass (?). That can make a big difference.
Did you use the same seeds for all beds? There is such a sharp sudden difference in the height of the plants it makes me wonder if they are two different varieties. Dwarf cosmos grow about 18-24 inches, regular ones up to 72 inches. Overcrowding will stunt growth, because there are too many roots beneath the soil competing for the nutrients and the plants above are all competing for air and sunlight and room to spread out. I would thin both patches by one-third to one-half. Also, if you have humid, damp weather, overcrowding is an invitation to fungus and mildew. The roots are are too close to pull them up, so use garden clippers and cut them off at the base. I know it sounds contradictory that you'll get more flowers with fewer plants, but it's true. (Plus I know it's hard to do, mentally, after you grew them all from seed. Ouch
) All of them will be happier and healthier with more room. Then go ahead and use whatever balanced fertilizer you decide on. I hope they give you lots of flowers.
