The Higgs boson decays to pairs of $b$-quarks were measured in associated production with a $W$ or $Z$ boson by the ATLAS collaboration at the Large Hadron Collider (LHC). The decay to $b$-quarks is of particular importance since it allows a direct measurement of the coupling of the Higgs boson to $b$-quarks. The highest sensitivity in this channel is obtained when the vector boson produced alongside the Higgs boson decays to leptons. The analysed data were collected in proton-proton collisions at the LHC during Run-2 at a center-of-mass energy of 13 TeV. The final state requires having exactly 2 $b$-tagged jets and either 0, 1 or 2 charged leptons (electrons or muons) corresponding to the following channels: $Z \rightarrow \nu \nu$, $W \rightarrow l \nu$ and $Z \rightarrow ll$. The analysis benefits from the full Run-2 data and from novel techniques used to improve the analysis sensitivity. The measurements yield the observation of the $VH$ signal as well as the observation of the $ZH$ signal and a strong evidence of the $WH$ signal. The results are validated with the diboson ($VZ$) analysis by measuring the $VZ, Z\rightarrow b \bar b$ signal strength, and are cross-checked with an analysis using the di-jet mass rather than the BDT$_{VH}$ output as main discriminant, which also leads to the observation of the $VH$ signal. All the results are in good agreement with the Standard Model expectations.