We present a study of the normalized transverse momentum distribution of $W$ bosons produced in $p \bar p$ collisions, using data corresponding to an integrated luminosity of 4.35 fb$^{-1}$ collected with the D0 detector at the Fermilab Tevatron collider at $\sqrt{s}=1.96$ TeV. The measurement focuses on the transverse momentum region below 15 GeV, which is of special interest for electroweak precision measurements; it relies on the same detector calibration methods which were used for the precision measurement of the $W$ boson mass. The measured distribution is compared to different QCD predictions and a procedure is given to allow the comparison of any further theoretical models to the D0 data.