A series of random growth models has been studied in which the growth probability at position r on the surface of the growing cluster is given by P(r)\ensuremath{\simeq}\ensuremath{\mu}(r)${\mathit{l}}^{\mathrm{\ensuremath{\varphi}}}$, where \ensuremath{\mu}(r) is the harmonic measure at r and l is the distance from the seed or origin. The distance l can be either the Pythagorean distance or the minimum path distance measured on the growing cluster. The introduction of this scale-invariant perturbation of the usual diffusion-limited-aggregation (DLA) model (\ensuremath{\varphi}=0) introduces a distance-dependent correlation length, \ensuremath{\xi}=l/\ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\Vert}, that characterizes a geometrical crossover in the cluster structure. Although the structures generated by these models have an appearance that is quite different from that of DLA clusters (for \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\Vert}\ensuremath{\gg}0), the growth of their radii of gyration and the internal density profile \ensuremath{\rho}(r) have simple power-law forms with the same exponents as those associated with DLA. The difference in scaling is manifest in the amplitudes of the power-law forms. These amplitudes exhibit a power-law dependence on the radial bias exponent \ensuremath{\varphi}.For \ensuremath{\varphi}\ensuremath{\gg}1 the clusters become self-affine structures with the same exponents as those associated with DLA on length scales r\ensuremath{\ll}\ensuremath{\xi}. These clusters exhibit a crossover to self-affine wedgelike linearly growing structures at r\ensuremath{\simeq}${\mathit{scrR}}_{\mathit{x}}$\ensuremath{\simeq}\ensuremath{\varphi}. For \ensuremath{\varphi}\ensuremath{\ll}-1 the growth probability is enhanced in the core of the clusters. These clusters exhibit a dense core having radius ${\mathit{scrR}}_{\mathit{x}}$\ensuremath{\sim}\ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\Vert}. For r\ensuremath{\simeq}${\mathit{scrR}}_{\mathit{x}}$, the structure crosses over to a structure having the same scaling behavior as DLA. For growth from a line in a strip of width L, the density-density correlation function in the lateral direction can be represented by the scaling form ${\mathit{C}}_{\mathit{h}}$(x)\ensuremath{\sim}${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\alpha}}}$g(x/${\ensuremath{\xi}}^{\mathrm{\ensuremath{\alpha}}/\ensuremath{\nu}}$), where h is the distance from the line substrate (height) and exponents \ensuremath{\alpha} and \ensuremath{\nu} have values of about 1/3 and 1/2, respectively. The scaling function g(x) has the form g(x)\ensuremath{\simeq}${\mathit{x}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ for x\ensuremath{\ll}1 and g(x)\ensuremath{\sim}const for x\ensuremath{\gg}1.