Ah yes, Khwarizm—here it is, the area around Khiva and—well, pretty much the territory in Uzbekistan where I was traveling last week. Muhammed succeeded his father about 1210. Over the next seven years, more or less, he pasted together an empire of sorts that extended out into Afghanistan and Persia. It was impressive on the map, although on close scrutiny, it appears to have been pretty fragile. Muhammed also seems to have had a knack for irritating near neighbors who might otherwise have become his allies.
At this point, Muhammed may never even have heard of his great contemporary, Genghis Khan, for it is a remarkable fact that Genghis, as a world leader, got late start. He didn't really consolidate among the Mongols until he was around 40, i.e., around 1207. He spent the next several years embroiled with the Chinese. It was only after that that he turned his attention to the west and found himself face to face with Muhammed.
Genghis' first approach seems to have been benign. He came looking for trade. But in 1218, a Khwazmarian governor robbed a Mongol caravan and put 100 of its members to death. Genghis was not amused. He demanded compensation and, being refused, made ready for war.* By the summer of 1219, Mongol forces began popping up all over Central Asia. In February of 1220, Genghis himself destroyed Bukhara. He moved on to loot Samarkand, reducing its population by about three quarters. In April of 1221, the Mongols finally took the old Khwarizm capital of Gurganj, modern Urgench: they drowned it by rerouting a river.
The rest, as they say, is history. The Mongols on into India and Iran. Genghis himself lived only until 1227—by a narrow reading, his entire career on the world stage lasted not much longer than seven years. But his sons and grandsons dominated China, Persia, Central Asia and the hinterland of “Russia” for generations to come.
*I'll say. "[T]he whirlwind of anger cast dust into the eyes of patience and clemency while the fire of wrath flared up with such a flame that it drove the water from his eyes and could be quenched only by the shedding of blood."--The Persian chronicler Ata-Malik Juvaini, quoted in Jack Weatherford, Genghis Khan and the Making of the Modern World at 107 (2004).