Let \(\mathbb{T}\) be the one dimensional torus represented by the interval \([-\pi,\pi]\) with the end points \(-\pi\) and \(\pi\) identified. For \(\mathbf{p}=(p_1,p_2)\) (\(1\leq p_1,p_2\leq +\infty\)), \(\mathbf{\rho}=(\rho_+,\rho_-)\) (a pair of nonnegative functions which may assume infinite values) and \(\psi(u,v)\) (an arbitrary asymmetric and monotone function, such that \(\Psi(u,v)\) is a norm on the plane), the author considers the asymmetric norm defined by \[ \psi_{\mathbf{\rho,p}}(f)=\psi\big(\| \rho_+(\cdot)f^{+}(\cdot)\| _{L_{p_1}(\mathbb{T})}, \| \rho_-(\cdot)f^{-}(\cdot)\| _{L_{p_2}(\mathbb{T})}\big), \] where \(f^+=\max\{f,0\}\) and \(f^-=\max\{-f,0\}\). Let \(L_{\psi_{\mathbf{\rho,p}}}(\mathbb{T})\) be the cone of the \(2\pi\)-periodic Lebesgue measurable functions \(f\) on \(\mathbb{T}\) such that \(\psi_{\mathbf{\rho,p}}(f)<+\infty\) and denote by \(\mathcal{T}_n\), \(n\in\mathbb{N}\), the set of all trigonometric polynomials of degree at most \(n\). For \(r\in\mathbb{N}\), let \(\text{BW}_{\psi_{\mathbf{\rho,p}}}^r(\mathbb{T})\) be the class of all functions \(f\) that have \((r-1)\)th derivative which is absolutely continuous on \(\mathbb{T}\) and \(f^{(r)}\in L_{\psi_{\mathbf{\rho,p}}}(\mathbb{T})\). The main theorem of this paper states that if \(\mathbf{p}=(p_1,p_2)\), \(\mathbf{q}=(q_1,q_2)\), with \(1\leq p_1,p_2,q_1,q_2\leq \infty\), \(\rho=(\rho_+,\rho_-)\), \(\widetilde{\rho}=(\widetilde{\rho}_+,\widetilde{\rho}_-)\) are two pairs of nonnegative functions and \(\psi\) is an asymmetric and monotone norm on the plane such that \(\psi(u,v)=\psi(v,u)\), then \[ \begin{aligned} E_n\Big(\text{BW}_{\psi_{\mathbf{\rho,p}}}^r(\mathbb{T}), L_{\psi_{\mathbf{\widetilde{\rho},q}}}(\mathbb{T})\Big):= &\sup_{f\in\text{BW}_{\psi_{\mathbf{\rho,p}}}^r(\mathbb{T})} E_n\Big(f,L_{\psi_{\mathbf{\widetilde{\rho},q}}}(\mathbb{T})\Big):= \\ &\sup_{f\in\text{BW}_{\psi_{\mathbf{\rho,p}}}^r(\mathbb{T})} \inf_{t\in\mathcal{T}_n}\psi_{\mathbf{\widetilde{\rho},q}}(f-t) \leq 8K_{r}(n+1)^{-r}C_\mathbf{\rho,\widetilde{\rho},p,q}(n).\end{aligned} \] where \(K_r\) is the so-called Favard constant and \(C_\mathbf{\rho,\widetilde{\rho},p,q}(n)\) are constants. Also, in some cases of asymmetric norms with weights \(\rho=(\alpha,\beta)\), where \(\alpha\) and \(\beta\) are real numbers in \([1,+\infty)\), the author finds the rate of decrease of \(E_n(\text{BW}_{\psi_{\mathbf{\rho,p}}}^r(\mathbb{T}), L_{\psi_{\mathbf{\widetilde{\rho},q}}}(\mathbb{T}))\) as \(n\to\infty\) for a fixed number \(r\in\mathbb{N}\).