The shapes of jets with transverse energies, $E_{\rm T}^{\rm jet}$ , up to 45 GeV produced in neutral- and charged-current deep inelastic $e^+p$ scattering (DIS) at $Q^2 > 100$ GeV $^2$ have been measured with the ZEUS detector at HERA. Jets are identified using a cone algorithm in the $\eta$ - $\phi$ plane with a cone radius of one unit. The jets become narrower as $E^{\rm jet}_{\rm T}$ increases. The jet shapes in neutral- and charged-current DIS are found to be very similar. The jets in neutral-current DIS are narrower than those in resolved processes in photoproduction and closer to those in direct-photon processes for the same ranges in $E_{\rm T}^{\rm jet}$ and jet pseudorapidity. The jet shapes in DIS are observed to be similar to those in $e^+e^-$ interactions and narrower than those in $\bar{p}p$ collisions for comparable $E_{\rm T}^{\rm jet}$ . Since the jets in $e^+e^-$ interactions and $e^+p$ DIS are predominantly quark initiated in both cases, the similarity in the jet shapes indicates that the pattern of QCD radiation within a quark jet is to a large extent independent of the hard scattering process in these reactions.