Watermelon vines often thin themselves to some degree, but in many cases, removing excess fruit can result in a better crop.

If the vines flower early, before they produce significant foliage (if I understand your post, this doesn't sound like your vine's problem), you can remove those early flowers to delay fruit set and encourage extra growth (needed to support the fruit later on).

Once the vines start producing fruit, wait until the fruit is about golfball size before removing it. Removing smaller fruit (before the vine aborts what it considers excess fruit) may be counter-productive. The number of melons a vine can support will vary with cultivar, weather, etc. So, leave your vine alone for a little while until it sets fruit and the fruit gets golfball size. Then, remove some fruit if you think there are too many melons.