Thing is, these carcinogens don't increase the odds of cancer all that much.
Massive percentage increases of tiny amounts produces...
Tiny amounts.
'Course, if you're eating carcinogens, smoking, getting X-rays every few weeks and so on, you're gonna have your odds increased to actually appreciable levels.
There's even more to it than that. Your cells aren't defenseless, there is a number of different mechanisms for fixing the DNA damage, and it would be strange if there wasn't - your DNA is being attacked by mutagens pretty much all the time, probably most commonly by light (or more accurately UV radiation). In small amounts, you can handle carcinogens pretty well.
The thing is, if your organism would fix 999 out of 1000 mutations back to the original state with no errors, if you splash it with 1000 there still be one mutation missed. Which most of the time is not that big an issue, most of the time, but it adds over time, and you can be unlucky and get the defenses damaged, increasing the likelihood of damage later on.
On top of that, add the fact that while DNA is pretty stable thanks to those, other components are far less so. RNA does not have any error-checking enzymes, nor does mitochondrial DNA, and thus your defense erodes with age, so amounts you could shrug off at the age of 18 might cause mutations at 80.