Chinese wheeled transport was dominated by 2 wheeled and 1 wheeled vehicles, but there were some 4 wheeled vehicles.

4 wheeled vehicles appear early on the Western Steppe, and were present in the 4th millenium BC. They appear (as grave finds) as far east as the Caspian Sea. They don't appear to have entirely vanished from the steppe; ibn Battuta reports the use of 4 wheeled wagons in the western steppe in the 14th century. Thus, there was use of such wagons near China.

The earliest Chinese depictions of 4 wheeled vehicles I know of are military. The Song Dynasty military compendium, Wujing Zongyao 武經總要, c. 1040, shows 4, 6, and 8 wheeled military equipment such as scaling ladders, artillery, siege towers, assault vehicles, and more (and 2 wheeled ones as well). The late Ming Wubei Zhi 武備志, 1621, also included more-than-2 wheeled vehicles, such as this siege tower.

The Ming Dynasty painting Departure Herald, 16th century, shows a large 4 wheeled mobile palace (near the right hand end).

The 18th century Qing version of Along the River During the Qingming Festival shows multiple 4 wheeled vehicles. These include a wagon carrying a large stone pulled by a large team of horses (a little to the left of the middle), some large 4 wheeled wagons with large loads pulled by 2 oxen; one is on the far side of the main bridge, and another behind some trees on the far side of the river a little to the left of that bridge. Just to the right of that last wagon is a passenger being pulled in a low 4 wheeled wagon (a toy wagon?). Almost all of the vehicles are 2 wheeled, 1 wheeled or no-wheeled; these 4 wheeled vehicles are a small minority. (The original Song Dynasty version has no 4 wheeled vehicles.)

Finally, the is an interesting 3 wheeled bronze vessel shown on pg 272 in Wen Fong et al. (eds), The Great Bronze Age of China, The Metropolitan Museum of Art, 2013.

Since 4 (and other more-than-2) wheeled vehicles were known and used, why did 2 wheeled and 1 wheeled vehicles dominate? 2 wheeled vehicles have many advantages. The main ones are cost (wheels are expensive), and turning ability. The worse the roads, the greater these two advantages are. When roads are poor, wheels should be large. Large wheels are more expensive, so the cost advantage of using 2 wheels instead of 4 is greater. Turning ability is a major advantage. With 2 wheels which can move independently on a fixed axle, the vehicle can turn on the spot. A 4 wheeled vehicle with two fixed axles is difficult to turn - it has a large turning circle. If one (or both) axles are pivoted, so that the axle can turn about a vertical axis, turning is simplified. However, the body of the wagon can block movement of the wheels. This can be prevented by having the body of the wagon above the wheels, but this requires either small wheels or a very high wagon body. Poor turning ability means that 4 wheeled vehicles can be difficult to operate in cities, compared to easy-to-turn 2 wheeled vehicles.

Richard Bulliet, The Wheel : Inventions and Reinventions, Columbia University Press, 2016 devotes a chapter to the Chinese lack of wider adoption of the 4 wheeled wagon. He notes that with the exception of the Ming portable palace in Departure Herald, the other examples are heavy-load wagons (and one toy). (We can also add military use.) The high status passenger vehicle was, except for the occasional mobile palace, 2, 1, or no-wheeled.