This paper studies boundedness and convergence of discontinuous Galerkin (DG(\(p\))) solutions of polynomials of degree \(p\) for general nonlinear scalar conservation laws with boundary conditions, and is an extension of J. Jaffre’s et al. work [Math. Models Methods Appl. Sci. 5, No. 3, 367–386 (1995; Zbl 0834.65089)], in which the Cauchy problem was investigated. The authors prove \(L^\infty (L^\infty )\)-boundedness of a higher-order shock-capturing streamline-diffusion DG(\(p\)) solution (\(p\geq 0\)). Based on this result, convergence to the entropy process solution of the initial boundary value problem is shown. The analysis in this paper is based on arguments demonstrated in [A. Szepessy, RAIRO, Modélisation Math. Anal. Numér. 25, No. 6, 749–782 (1991; Zbl 0751.65061)], which are valid for \(p=1\), and on using the semi-Kružkov entropy pairs so that the discrete solution satisfies appropriate discrete entropy inequalities. The framework of the entropy process solution then ensures convergence of the DG(\(p\)) solution to the unique weak entropy solution.