There have been a few different shifts in attitude and law changes over the years. Not easy to find a picture of this with "a quick google" really.
Early 80s is when you really start to see child safety seats appear as a category e.g. in the old Argos catalogues, which suggests they were becoming more commonplace around that time. Fits with the first seatbelt laws in 1984. Previous to this, there was perhaps one option, presented as a speciality item. Or you could have child seats fitted by the car dealer if you were well off. Nothing much for babies, except straps to fix carrycots into the car, and a little hammock thing for the parcel shelf. The carrycot straps had no crash safety benefit, they just stopped it sliding around corners, but maybe there's a perception that there was crash safety. From anecdotes I think at this time it was very much social circle dependent - some people thought it was stupid and dangerous to drive kids around without car seats, and some people thought it was crazy OTT to bother with them. Even if you did use them you'd have to go without if you used another car because they were permanently fitted, not easy to move around.
1985 was the first rear facing baby seat (for the UK) - they came much later than forward facing ones because the design we have always used in the UK has to be fitted with seatbelts. They were also some of the first child seats to be able to be fitted with seatbelt. The first baby car seats were normally used in the front, because that was where the seatbelts were. Most of them didn't have handles - so they were probably quite impractical/expensive for what they were. I'd imagine most people carried on using carrycots because you'd have one for your pram anyway.
You could also get a "two way" seat which was also seatbelt fitted and was designed to be used backward facing with a recline up to 9kg (6-9 months) and then forward facing once they can sit up. I would guess these would have been more popular as better value (but honestly I don't know).
First car seat law came in in 1989:
All child passengers must use a car seat if a suitable one is fitted. If not:
Babies up to 1 can use a carrycot (strapped in)
Children 1-4 can use a seatbelt (but only with a booster seat)
Children 4-11 can use a seatbelt
If these options are not available, children could legally travel unrestrained.
I understand that all of this was meant to allow for people not having car seats permanently fitted in grandparents' cars, take their niece/nephew on a day out, give their neighbour or friend a lift etc. Likely because the seatbelt fitted seats were still new, and not all cars had rear seatbelts. But in practice, it meant that it was perfectly legal not to even own a car seat in the first place.
Early 90s: Big shift, many more models available, and seatbelt-fitting became common. Seatbelts were required in the rear of all cars from 1991, but older cars didn't have to retrofit them.
Also, the first proper car seat law in 1993: Children under 3 now had to use car seats (still an exception for babies in carrycots) - the only exception was where the car has no seatbelts at all. It does seem that by this point, car seats were a totally standard, normal thing to use so the legislation likely just cemented in law what was already typical practice.
By the mid 90s most infant seats had the carry handle, and were marketed as being a combination car seat/low chair/rocker - people fed their babies their first solids in car seats. They were seen as being really versatile. Not much commentary yet about the dangers of overusing car seats.
1995 - a couple of notable changes with the newest EU regulation R44.03 - Group 0+ was added so infant carriers could now go up to 13kg (not just 10kg) and a car certification was added for carrycots. Unfortunately I can't find the corresponding UK legislation so I am not sure when it changed from allowing any carrycot to be used to only certified ones. Airbag warning added to rear facing car seats, since this had become known as a danger, and many people were still using these seats in the front of the car.
Travel systems (buggy compatible car seats) were invented around 2000/2001-ish and these were HUGE - totally changed the way people chose both prams AND car seats, to the point they are basically seen as an essential today. Fun thing is that MN starts around this time, so if you use the advanced search you can go back and read actual conversations people were having about them when they were a totally new concept. (And be amazed by the prices people thought were expensive :D)
By the late 00s I had a baby myself, so I know that the advice had changed from "move them forward facing when they can sit up" to "wait until minimum 9kg, 9 months and as late as possible is better" (but still barely anyone kept them rear facing after 12 months) and it was definitely the norm to use a car seat rather than a car carrycot, because the car carrycots were so huge and bulky and most carrycots weren't certified.
2013 were the first seats sold with the "minimum 15 months to forward face" rule, although it took a long time for this to catch up to general practice. I would say we are just about there now.
Interesting to match up the changes in attitudes and legislation, especially through the 60s-00s, with the numbers here:
en.wikipedia.org/wiki/Reported_Road_Casualties_Great_Britain#/media/File:Killed_on_British_Roads.png