Measuring the properties of extragalactic magnetic fields through the effect of Faraday rotation provides a means to understand the origin and evolution of cosmic magnetism. Here, we use data from the LOFAR Two-Metre Sky Survey (LoTSS) to calculate the Faraday rotation measure (RM) of close pairs of extragalactic radio sources. By considering the RM difference (ΔRM) between physical pairs (e.g. double-lobed radio galaxies) and non-physical pairs (i.e. close projected sources on the sky), we statistically isolate the contribution of extragalactic magnetic fields to ΔRM along the line of sight between non-physical pairs. From our analysis, we find no significant difference between the ΔRM distributions of the physical and non-physical pairs, limiting the excess Faraday rotation contribution to <1.9 rad m−2 (∼95 per cent confidence). We use this limit with a simple model of an inhomogeneous universe to place an upper limit of 4 nG on the cosmological co-moving magnetic field strength on Mpc scales. We also compare the RM data with a more realistic suite of cosmological magnetohydrodynamical simulations that explore different magnetogenesis scenarios. Both magnetization of the large-scale structure by astrophysical processes such as galactic and AGN outflows, and simple primordial scenarios with seed magnetic field strengths <0.5 nG cannot be rejected by the current data; while stronger primordial fields or models with dynamo amplification in filaments are disfavoured.