This paper presents a dedicated search for exotic decays of the Higgs boson to a pair of new spin-zero particles, $H \rightarrow aa$, where the particle $a$ decays to $b$-quarks and has a mass in the range of 20-60 GeV. The search is performed in events where the Higgs boson is produced in association with a $W$ boson, giving rise to a signature of a lepton (electron or muon), missing transverse momentum, and multiple jets from $b$-quark decays. The analysis is based on the full dataset of $pp$ collisions at $\sqrt{s} = 13$ TeV recorded in 2015 by the ATLAS detector at the CERN Large Hadron Collider, corresponding to an integrated luminosity of 3.2 fb$^{-1}$. No significant excess of events above the Standard Model prediction is observed, and a $95\%$ confidence-level upper limit is derived for the product of the production cross section for $pp \rightarrow WH$ times the branching ratio for the decay $H \rightarrow aa \rightarrow 4b$. The upper limit ranges from 6.2 pb for an $a$-boson mass $m_a = 20$ GeV to 1.5 pb for $m_a = 60$ GeV.

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