We obtained growth estimates for bivariate functions which are analytic in the unit bidisc and have bounded $\mathbf{L}$-index in joint variables.
The positive continuous function $\mathbf{L}(z_1,z_2)=(l_1(z_2,z_2),l_2(z_1,z_2))$ satisfies additional behavior condition:
for every point $z=(z_1,z_2)$ belonging the unit bidisc $\mathbb{D}^2$ the appropriate value of the function $l_j$ at this point is greater than the reciprocal to
$1-|z_j|$ multiplied by $\beta$, i.e. $l_j(z)>{\beta}/{(1-|z_j|)}$ for each $j\in\{1,2\}$
and some constant $\beta>1.$ Also  we prove that for every analytic functions in the unit bidisc with bounded multiplciities of zero points there exists a positive continuous function
$\mathbf{L}(z_1,z_2)=(l_1(z_2,z_2),l_2(z_1,z_2))$ providing boundedness of the $\mathbf{L}$-index in joint variables for primary analytic function.

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