The crash happened at 5300 South 400 West.
Emergency crews worked to get the women and children out of the SUV, that became lodged underneath sand/gravel trailer.
Police say the driver and her adult daughter were riding in the front seat of the vehicle, while two small children were riding in car seats in the back seat.
The driver's grandchildren were not seriously hurt but were taken to a local hospital as a precaution.
The driver and her daughter were eventually extricated and taken to a hospital.
Police say the drive lost control at the bottom of the northbound I-15 off ramp at 5300 South, where she crossed the median and crashed into the stopped truck.
Stay tuned to ABC 4 News and ABC4.com for more on this story.