The outcome of the Opium Wars (1839-1842 and 1856-1860) resulted in significant changes for China, primarily in the form of territorial and economic concessions to Western powers. The treaties that ended the wars, especially the Treaty of Nanking (1842) and the Convention of Peking (1860), forced China to cede Hong Kong to Britain, open up several ports to foreign trade, and grant extraterritorial rights to foreign nationals.
While the Opium Wars did contribute to social unrest in China and various subsequent rebellions, such as the Taiping and Boxer Rebellions, the statement about these rebellions being put down by Western powers is more accurate and contextually relevant. These unrests were a result of the social and political instability exacerbated by the conditions imposed after the Opium Wars, which led to significant Western influence in China.
To summarize, the most relevant outcome of the Opium Wars was the imposition of unequal treaties that weakened China's sovereignty and contributed to internal conflicts.