Did they? For the sake of argument, China went through 12 dynastic periods between 221 BC and AD 1912. Tabulated, these were:

Dynasty/period	Time	Full Years in Power
Qin	221-206 BC	14
Western Han	202 BC-AD 9	220
Eastern Han	25-220	214
Three Kingdoms	220-280	60
Jin	265-420	154
Northern and Southern	420-589	169
Sui	581-618	37
Tang	618-907	289
5 Dynasties & 10 Kingdoms	907-979	72
N. Song	960-1127	166
S. Song	1127-1271	143
Yuan	1271-1368	96
Ming	1368-1644	275
Qing	1644-1912	268

^(Italic text indicates this was not a period with one main ruling dynasty)

As you can see, a typical dynasty lasted around 100 years at least, with the exception of the brief Qin and Sui periods, and indeed often exceeded 200. For a counter-point, look at France: the Merovingians, Carolingians (excluding various interregna), Valois and Bourbons lasted around 250 years each on the throne, and only the mainline Capetians exceeded the 300-year mark. While a bit longer, it's not exceptionally so – and I have opted to ingore several French interregna here. And if we're calling China exceptionally unstable, well while the Qing remained continuously on the Chinese throne between 1644 and 1912, France was on a political rollercoaster from 1789 onwards, with the revolution, the rise of Bonaparte, the Bourbon Restoration, the 1848 revolution, the establishment of the Second Empire, and the establishment of the Third Republic.

Rome is even worse, and no dynasty ruled the united empire for more than 100 years, even with the loose succession rules of the Julio-Claudian period. The closest they got was the Nerva-Antonines, who ruled from 96 to 192. There were only three centennial Byzantine dynasties which were the Macedonian (Basil I through Constantine IX) and Komnenid (Alexios I through Andronikos I), and only barely so, with the Palaiologoi standing out as the most long-lasting, spending 191 full years on the throne between 1261 and the fall of Constantinople to the Ottmans in 1453.

So I'm afraid I can't answer your question because its premise is flawed. China has just not been an exceptionally unstable place in terms of dynastic succession relative to any European dynasties you may be thinking about.

This question makes assumptions I don't feel are entirely accurate. First, the idea that they fall apart "so often" is a subjective statement and ignores that while there are short-lasting dynasties such as the Qin dynasty, there were also long-lasting dynasties such as the Tang dynasty. Additionally, the historical context and conditions which each dynasty and ruler found themselves in make it hard to form an accurate, but broad enough generalised reason that it would cover every dynastic fall.

Additionally, your question, to me, sounds like you're treating dynasties as completely separate countries. While there's a lot of criticism given to the whole "China is forever continuous, etc." claim, the opposite is equally wrong. Just because a new dynasty has formed does not mean that it is a completely separate civilisation from the previous, or that it doesn't have continuity with the previous dynasty. If you compare a single dynasty to the history of France for example, then of course a Chinese dynasty/nation looks remarkably prone to disaster. However, if you look at the dynastic history of France, it went from the Capets, to the Valois, and then the Bourbon. In this same time period, China went from Song/Jin, to the Yuan, to the Ming, and then the Qing. England in comparison looks even more chaotic from a dynastic point of view: England went from frequent wars between Scandinavian and Anglo-Saxon claimants to the house of Normandy, to the Blois (short-lived), then the Anjou, Plantegenet, then the War of the Roses debacle and the Lancasters/York, then the Tudors, Stuarts, and then finally the German houses (Hanovers, Saxe-Coburg and Gotha/Windsor); looking at it from this point of view, China looks remarkably stable in terms of dynastic change! At the same time, I'm not denying that Chinese dynasties differ significantly from their European counterparts. For example, Chinese dynasties rarely ran out of heirs because of the power of polygamy (:D) while monogamy meant that European dynasties could and often did struggle with producing enough heirs. This also meant that there was a fundamental difference in how Chinese dynasties could change vs. European dynasties. European dynastic transitions could be peaceful as a result of a female heir; the Stuart dynasty ended with Anne, but was succeeded peacefully by the Hanoverians, who themselves (after Victoria) would be peacefully succeeded by the house of Saxe-Coburg and Gotha. A comparable succession really doesn't exist for Chinese dynasties. Hence, a one-to-one comparison is difficult to make in any case.

You also have to remember that dynasties didn't conveniently form over one night. While there might be a traditional date marking the formation of a dynasty, it's not like the dynasty was functionally non-existent prior to the date. Take the Sui dynasty for example: traditionally, the formation of the dynasty dates to 581 AD, which is after Emperor Wen of Sui overthrew the Northern Zhou and established his own dynasty. However, by this point in time, Northern Zhou was the dominant power within the Chinese region and had defeated the remaining Chen dynasty in battle prior to their own collapse. The establishment of Sui was thus, top-down rather than bottom-up and to view Northern Zhou's transition to Sui without recognising some form of continuity feels unfair. At the same time, Northern Zhou didn't come up suddenly; it was the last dynasty in a line preceded by the Western and Northern Wei, which extends itself all the way back to the 4th century. While the dynasties and the states themselves changed over the centuries, it's foolish to imagine a simple line to draw that can state when a state fell apart and when it formed again, with dynastic periodisation being a terrible and inaccurate line to draw. Even after the overthrow of the Sui, it wasn't as if the Tang appeared out of nowhere. The Sui dynasty by the time of its nadir was overstretched and was recovering from a disastrous war with the Goguryeo. The Tang dynasty was borne out of many rebellions against the Sui dynasty in their last years and would finish integrating/conquering the various territories that were part of the Sui dynasty over the course of a decade. While the Tang dynasty itself would change and morph over the following centuries, the Tang dynasty at the time of its conquest and reunification of China was quite similar to the Sui dynasty that preceded it in both governance, culture, territory, etc. So did the Northern Zhou, then the Sui, and eventually the Tang all fall apart? Of course, but did that make the entities they governed entirely separate countries? Rather disputable.

I guess what I'm trying to convey is that you can't think of Chinese dynasties as conveniently separate countries, and that European countries in comparison were subject to frequent wars and instability as well. There's nothing special about the dissolution and reformation of China as an entity over the years, other than maybe its size. They didn't fall apart faster than any other civilisations did, and how we choose to interpret and view historical events can change how we view the continuity of a state or a civilisation.