There must be some high-mathematical orbital dynamics reason for that, but yes, they must be only a few km wide.
Have you seen Cassini pictures of the moonlets that make the gaps in the rings? There's one (and a movie) that shows waves in the rings created by the moon as it goes by. Wild.
For a few, not even kilometers thick.
http://solarsystem.nasa.gov/planets/profile.cfm?Object=Saturn&Display=Rings
Others are thicker: probably due to the moons being closer.
Ring Name: D
Distance*: 68,000 km
Width: 8,500 km
Ring Name: C
Distance*: 74,500 km
Width: 17,500 km
Mass: 1.1 x 1021 kg
Albedo: 0.1-0.3
Ring Name: B
Distance*: 92,000 km
Width: 25,500 km
Thickness: 0.1 km - 1 km
Mass: 2.8 x 1022 kg
Albedo: 0.4-0.6
Ring Name: Cassini Division
Distance*: 117,500 km
Width: 4,700 km
Mass: 5.7 x 1017 kg
Albedo: 0.2-0.4
Ring Name: A
Distance*: 122,200 km
Width: 14,600 km
Thickness: 0.1 km - 1 km
Mass: 6.2 x 1021 kg
Albedo: 0.4-0.6
Ring Name: F
Distance*: 140,210 km
Width: 30 km - 500 km
Albedo: 0.6
Ring Name: G
Distance*: 165,800 km
Width: 8,000 km
Thickness: 100 km - 1,000 km
Mass: 1 x 1017 kg
Ring Name: E
Distance*: 180,000 km
Width: 300,000 km
Thickness: 1,000 km - 30,000 km
Mass: 7 x 108 kg
* The distance is measured from the planet center to the start of the ring.