The numerical solutions of stochastic differential delay equations (SDDEs) under the generalized Khasminskii-type condition were discussed by Mao (2011) and the theory there showed that the Euler--Maruyama (EM) numerical solutions converge to the true solutions \emph{in probability}. However, there is so far no result on the strong convergence (namely in $L^p$) of the numerical solutions for the SDDEs under this generalized condition. In this paper, we will use the truncated EM method developed by Mao \cite{M15} to study the strong convergence of the numerical solutions for the SDDEs under the generalized Khasminskii-type condition.