IMO, too many people saying to go too big.
Let the terrain dictate height.
The culvert as a whole cannot be dug deeper than the outlet. So the "bottom" is gonna be fixed in the same location weather you use a 4" or a 36" culvert. So let the height be the deciding factor. A 18" culvert with 9" of dirt on top is 27" from the fixed "bottom" point. If that is significantly higher than the rest of the driveway, its gonna leave a hump, create a speed bump, and just look like crap IMO.
So look at the rest of the drive and determin the maximum height that would be acceptable. Size accordingly assuming 50% fill on top. Then if you need more flow, add a second or third culvert.
Oh, and adding 50% to the diameter( in the example above going from 12 to 18) only adds about double the capacity. (2.25 times to be exact). Not 5x's. Doubling diameter gets you 4x's the volume.
Of course all this assuming that the culvert getting crushed was the issue to begin with. Maybe I missed it but didn't see plugging as the primary issue