Some time ago I did do some research on car weights. I think these were mostly published 'curb' weights of the basic models.
Top ten best selling cars in 1979 were:
Ford Cortina: 1110kg
Ford Escort: 767kg
Mini: 675kg
Morris Marina: 885kg
Austin Allegro: 869kg
Ford Fiesta: 771kg
Ford Granada: 1190kg
Ford Capri: 1010kg
Vauxhall Cavalier: 895kg
Vauxhall Chevette: 845kg
In 2013:
Ford Fiesta: 1150kg
Ford Focus: 1270kg
Vauxhall Corsa: 1086kg
Vauxhall: Astra: 1200kg (my est.)
VW Golf: 1346kg
Nissan Qashqai: 1297kg
BMW 3 series: 1495kg
VW Polo: 1212kg
BMW 1 Series: 1360kg
Peugeot 208: 1168kg
So between 1979 and 2013 an overall increase in weight of 39.6%. Presumably they've only got even heavier in the last 6 years.
The direct comparisons there are:
Ford Escort: 767kg - Ford Focus: 1270kg
Ford Fiesta: 771kg - Ford Fiesta: 1150kg
Vauxhall Chevette: 845kg - Vauxhall: Astra: 1200kg (my est.)
Part of the problem here is that they are cars which had to put it mildly, diabolical safety considerations.
An ideal comparison would be to take the 1970s SAAB or Volvo and compare it to now... Since SAAB no longer exist we're left with comparing Volvos
But ok a quick look at SAABs
IIRC the late model SAAB 99s had a plate weight of 1200kg and the mid-80s 900s were around 1400kg
The weights listed for the final 2014 NEVS 9-3 are 1410 to 1690kg
Not such a big increase in weight where the old side of the comparison had similarly thick doors with Side Impact Bars in them (SAAB introduced them mid-70s IIRC along with dashboards that offered some padding, A-pillars that were padded and pretty strong and weren't creating huge blind spots thanks to the front window having an epic curve in them to allow it).
Wikipaedia is helping with the Volvos giving a timeline of direct replacements
1970/80s 200 Series - 1270kg - 1465kg
1990s 850 Series - 1385kg - 1570kg
Late 90s to 2016 V70 - 1410kg to 1518Kg (FWD, 4WD adds ~300kg)
The V90 is given as the successor but it's clearly not a direct replacement
So while Safety features aren't the only cause of weight increase, they are clearly the significant portion of it, and when manufacturers who previously thought it fit to sell cheap death traps made of tin were forced to stop doing that of course the average weight increased.