My conclusion after observing the two words is that they are not related.
The right component of 龍 depicts the form of a Loong with spikes on its back; while 弱 seems to relate to decorative features like 羽 (feather)
The character 弱(weak) came from the image of a decorated bow, which implies it is not as strong as a practical bow
Reference
弱: deogrammic compound (會意/会意): 弓 (“bow”) + 彡 (“hair, meaning "decoration" in this case”) - a decorative bow that is functionally inferior. Or it might be indicating that the bow is shaking, which implies a worn-out bow.
龍: Current form developed in large seal script, with serpent’s body on right (tail at upper right, legs on right)