Well, after asking the question about the offending Company and checking some of the Jeep Safaris out for myself, we went on the safari after all with Karacay (or a similar name to that).
Big jeeps, big comfortable seats, seatbelts, everybody was strapped in and the jeeps had full roll bars (I know cos I banged my head on them getting in and out on a number of occasions. We all felt pretty safe in the jeep, though there is plenty of wind in the hair and water being hosed over you by kids in roadside locations en route.
As most of the jeeps follow a similar route, I saw a number of the smaller jeeps, still with people standing, and the sitting ones looking cramped. Also the seating positions seemed high against the side meaning any sudden sideways movement looked like the occupants could fall out. I have to say I was pretty glad with the choice of Company I made. And we did have an excellent day.