4 possible causes:
1. Cold damage. Since you recently got it, I'm assuming it started growing some time in winter. Though not likely, some vascular tubes may have been damaged and it can't bring up enough water and nutrients to certain portion of the plant, causing it to turn brown once it reached certain size.
2. Not enough sun. Since you keep it indoor, it could be that it's not getting enough sun, which in turn causes lack of hormones produced, causing photosynthesis to be less effective.
3. Too much sun. Even if you keep it indoor, now that sun's getting hotter, it could be simply burnt. Do you rotate your plant regularly? If you don't, and if the brown spots are where the plant is facing the window, it could be just burnt.
4. Too much water. Since you keep it well watered, I don't think it's lack of water, but it could be too much water. You should water it to the point where water comes out at the bottom of the pot, but remove the excessive water afterward. Too much water can cause roots to rot, causing water to be not absorbed.
Mine gets brown over the winter, mostly due to not enough sun. I keep it outside once it's 10-15c outside, but bring it in over night if it goes below 5c. In spring, I keep it on the south side of the house so it gets plenty sun, but once it gets hot (around early June), I bring it to the east side of the house, because the sun's just too hot. I usually trim off the brown part, and it grows new branches over the summer, making it full again.
Once it reaches certain height you desire, cut off the top. Once you cut off the top where the meristem is (google Shoot Apical Meristem if you need to identify the part), it won't get any taller, but it'll send all the nutrients to the rest of the plant, making it leafier. With SAM intact, plant will keep growing upward, using up most of nutrients for growing taller till the plant reaches its natural height. But once SAM's gone, all the nutrients will make it leafier. That's why most indoor weed are short and leafier (because people cut off SAM once they reach 3-4' tall) compared to outdoor weed that are just tall and not leafy at all.
And, to keep the topic in track, I present to you, eggplant.
