In the worst cases with the small cars that did not score well, safety cages collapsed, driver airbags moved sideways and the crash dummy's head hit the instrument panel, and side curtain airbags did not deploy or provide enough protection, IIHS chief research officer David Zuby said.
The small car segment was the fourth group of cars rated using this new test, and most of the groups have fared equally badly.
Last summer, seven of 11 luxury sedans evaluated rated "marginal" or "poor," and 12 of 15 small SUVs tested also failed to score well in results released in May. Family sedans scored the best, with only five of 18 scoring "marginal" or "poor" in results released last December.
As a group, the small cars fared worse than the mid-sized family sedans, but better than the small SUVs, IIHS said. Results on the new crash test for minicars will be released later this year.
The other six small cars tested included two- and four-door versions of Honda Motor Co Ltd's Civic, which both received "good" ratings. The Civic was tested earlier this year and the results were released in March.
Receiving "acceptable" ratings were Chrysler's Dodge Dart, Ford Motor's Focus, Hyundai Motor's Elantra and Toyota Motor's 2014 Scion tC.
All the cars scoring well received "Top Safety Pick " ratings by the insurance trade group. Vehicles earning the institute's "Top Safety Pick " award have received "good" ratings in the four traditional tests plus "good" or "acceptable" ratings in the small overlap test.
IIHS said it did not test the Toyota Corolla because the automaker plans to release a redesigned 2014 model this month.