To estimate the effectiveness of the bivalent mRNA booster vaccines containing the original SARS-CoV-2 and omicron BA.4-5 or BA.1 subvariants as the fourth dose against severe covid-19. Design: Nationwide cohort analyses, using target trial emulation. Setting: Denmark, Finland, Norway, and Sweden, from 1 July 2022 to 10 April 2023. Participants: People aged ≥50 years who had received at least three doses of covid-19 vaccine (that is, a primary course and a first booster). Main outcome measures: The Kaplan-Meier estimator was used to compare the risk of hospital admission and death related to covid-19 in people who received a bivalent Comirnaty (Pfizer-BioNTech) or Spikevax (Moderna) BA.4-5 or BA.1 mRNA booster vaccine as a fourth dose (second booster) with three dose (first booster) vaccinated people and between four dose vaccinated people. Results: A total of 1 634 199 people receiving bivalent BA.4-5 fourth dose booster and 1 042 124 receiving bivalent BA.1 fourth dose booster across the four Nordic countries were included. Receipt of a bivalent BA.4-5 booster as a fourth dose was associated with a comparative vaccine effectiveness against admission to hospital with covid-19 of 67.8% (95% confidence interval 63.1% to 72.5%) and a risk difference of -91.9 (95% confidence interval -152.4 to -31.4) per 100 000 people at three months of follow-up compared with having received three doses of vaccine (289 v 893 events). The corresponding comparative vaccine effectiveness and risk difference for bivalent BA.1 boosters (332 v 977 events) were 65.8% (59.1% to 72.4%) and -112.9 (-179.6 to -46.2) per 100 000, respectively. Comparative vaccine effectiveness and risk difference against covid-19 related death were 69.8% (52.8% to 86.8%) and -34.1 (-40.1 to -28.2) per 100 000 for bivalent BA.4-5 booster (93 v 325 events) and 70.0% (50.3% to 89.7%) and -38.7 (-65.4 to -12.0) per 100 000 for BA.1 booster (86 v 286) as a fourth dose. Comparing bivalent BA.4-5 and BA.1 boosters as a fourth dose directly resulted in a three month comparative vaccine effectiveness and corresponding risk difference of -14.9% (-62.3% to 32.4%) and 10.0 (-14.4 to 34.4) per 100 000 people for admission to hospital with covid-19 (802 v 932 unweighted events) and -40.7% (-123.4% to 42.1%) and 8.1 (-3.3 to 19.4) per 100 000 for covid-19 related death (229 v 243 unweighted events). The comparative vaccine effectiveness did not differ across sex and age (</≥70 years) and seemed to be sustained up to six months from the day of vaccination with modest waning. Conclusion: Vaccination with bivalent BA.4-5 or BA.1 mRNA booster vaccines as a fourth dose was associated with reduced rates of covid-19 related hospital admission and death among adults aged ≥50 years. The protection afforded by the bivalent BA.4-5 and BA.1 boosters did not differ significantly when directly compared, and any potential difference would most likely be very small in absolute numbers.