In this paper, we propose a new weighted square-root LASSO procedure to estimate the regression coefficient matrix in sparse multivariate regression model with high-dimensional responses. The key advantage of the methodology is that it does not require the knowledge of the error term and has the tuning-insensitive property. To account for the within-subject correlation between responses, we use a working precision matrix which can be easily obtained in practice. Oracle inequalities of the estimators are derived. The performance of our proposed methodology is illustrated via extensive simulation studies. An application to DNA methylation data is also provided.