The relative positions of the high and low surface brightness regions of radio-loud active galaxies in the 3CR sample were found by Fanaroff and Riley to be correlated with their luminosity. We revisit this canonical relationship with a sample of 5805 extended radio-loud active galactic nuclei (AGN) from the LOFAR Two-Metre Sky Survey (LoTSS), compiling the most complete data set of radio-galaxy morphological information obtained to date. We demonstrate that, for this sample, radio luminosity does not reliably predict whether a source is edge-brightened (FRII) or centre-brightened (FRI). We highlight a large population of low-luminosity FRIIs, extending three orders of magnitude below the traditional FR break, and demonstrate that their host galaxies are on average systematically fainter than those of high-luminosity FRIIs and of FRIs matched in luminosity. This result supports the jet power/environment paradigm for the FR break: low-power jets may remain undisrupted and form hotspots in lower mass hosts. We also find substantial populations that appear physically distinct from the traditional FR classes, including candidate restarting sources and ‘hybrids’. We identify 459 bent-tailed sources, which we find to have a significantly higher SDSS cluster association fraction (at z < 0.4) than the general radio-galaxy population, similar to the results of previous work. The complexity of the LoTSS faint, extended radio sources not only demonstrates the need for caution in the automated classification and interpretation of extended sources in modern radio surveys, but also reveals the wealth of morphological information such surveys will provide and its value for advancing our physical understanding of radio-loud AGN.