A supermoon does not occur every three or four months. There may be 2 or 3 consecutive supermoons (that is, separated by 1 month) that occur at about the same time each year.
If you ignore the precession of the Moon's orbit, then there is one time of the year when the Full Moon and perigee occur (point 1 in my diagram below). Two weeks later, the New Moon occurs near apogee (point 2). At any other time of the year, the Full Moon occurs at a different point in its orbit around the Earth, so it is farther from the Earth than at perigee. Six months later, the Full Moon is occurring at apogee (point 4), and no one cares about that!
(not to scale!)
- Full Moon at perigee (closest to the Earth)
- New Moon at apogee (farthest from the Earth)
- New Moon at perigee
- Full Moon at apogee
The Wikipedia article on the Supermoon has a nice graphic showing the Full Moon and distance from the Earth. Depending on how close to the Earth the Moon needs to be to be "super", you can see that there is a "season" when the supermoon occurs. (I have copied the image here, and added a dashed line at 360,000 km to show which moons might be "super" and which ones are not.)
Now, if only people would care about the Super First Quarter Moon. Then we would be celebrating on May 12, 2019! (Not really. That is farther than the 360,000 km criteria.)