  The method of calculating electroweak decay matrix elements between
heavy-heavy and heavy-light meson states is developed in the framework of
relativistic quark model based on the quasipotential approach in quantum field
theory. This method is applied for the study of exclusive semileptonic
$B\to\pi(\rho)$ decays. It is shown that the large value of the final
$\pi(\rho)$ meson recoil momentum allows for the expansion in inverse powers of
$b$-quark mass of the decay form factors at $q^2=0$, where $q^2$ is a momentum
carried by the lepton pair. This $1/m_b$ expansion considerably simplifies the
analysis of these decays and is carried out up to the second order. The
$q^2$-dependence of the form factors is investigated. It is found that the
$q^2$-behaviour of the axial form factor $A_1$ is different from the other form
factors. It is argued that the ratios $\Gamma(B\to\rho e\nu)/\Gamma(B\to\pi
e\nu)$ and $\Gamma_L/\Gamma_T$ are sensitive probes of the $A_1$
$q^2$-dependence, and thus their experimental measurement may discriminate
between different approaches. We find $\Gamma(B\to\pi e\nu)=(3.0\pm0.6)
\times|V_{ub}|^2\times10^{12}$s$^{-1}$ and $\Gamma(B\to\rho e\nu) =(5.4\pm
1.2)\times|V_{ub}|^2\times10^{12}$s$^{-1}$. The relation between semileptonic
and rare radiative $B$-decays is discussed.
