Watermelon vines are highly prone to viruses, bacterial and fungal diseases. If you’ve grown Tomatoes, papaya, cucumbers or other plants susceptible to fungal diseases, you probably have a high bioburden which can infect your melon vines.
Looking at your picture, I would say that when your plant was smaller, you may have overwatered your plant. It appears, from the above ground vines and leaf size, that the roots may be weak and have not filled your pot. This often happens when the root zone is over saturated with water causing anaerobic conditions which are not conducive to root growth. The anaerobic conditions creates a good growing environment for the bad organisms.
To remedy this, be sure to start with good clean soil and preferably a watermelon variety that is disease resistant. Watermelon vines hate having their roots disturbed and they hate being transplanted but I’ve found that it’s best to start them off in smallish containers like a #1 container and let them completely fill the pot with roots before carefully transplanting them into their final container.
By doing it this way, you can control the soil moisture more easily and get your seedling in a strong healthy state before putting it into its final big pot. The roots should completely fill the #1 container before transplanting and this will ensure that it will have ample root mass to rapidly grow into the bigger container. Your plant should be given full sun and the shade from your nearby plants will slow down the growth of your vines. Once you transplant into the bigger container, water it thoroughly but let the soil dry out in between watering so the roots will spread out in search of water.
Simon