Searches for the decays of the Standard Model Higgs boson to dielectrons and to dimuons represent an important test of the yet unobserved Higgs boson couplings to first- and second-generation fermions. In addition, by looking for the Higgs boson lepton-flavour-violating processes such as $H\rightarrow e\mu$ or $H\rightarrow e\tau$ and $H\rightarrow \mu\tau$, which are forbidden in the Standard Model, it is possible to probe Beyond Standard Model theories allowing for such decays. The results of the presented searches are based on the partial and full Run-2 dataset, corresponding to an integrated luminosity of 36.1 fb$^{-1}$ and 139.0 fb$^{-1}$, respectively, collected by the ATLAS detector in $pp$ collisions at the LHC with the centre-of-mass energy of 13 TeV.