What causes airliner tyres to burst as often as they do?
Car tires aren't really comparable. 

Car tires: 


make a lot of miles, but in mostly benign circumstances. Low speeds mean little heating (and slow changes in temperature).
failures are often caused by debris in the road rather than structural failure of the tire itself. (Although on trucks, structural failure is more common)
must be replaced when the tire profile reaches a minimum depth, which means you can't keep using the tire until structural failure happens.
are generally thrown away when the profile reaches minimum depth, because car tires are cheap enough that retreading is not necessary


Airplane tires: 


rapidly go through temperature cycles (in-flight: -50 ºC, and on touchdown they rapidly heat up to several hundred ºC)
are kept at much higher pressures (13 bar versus 2 bar for car tires), so if there's a failure it's more likely to be catastrophic.
need to run at high speeds (300 km/h and more)
carry a lot of weight (up to 38 tons, vs. 500 kg for a car tire and 5 tons for a truck tire), which explains why they need such high pressures.
have to be lightweight
are often refurbished