Lots of things can lead to SharePoint 2013 performance issues, especially in the Azure cloud. While Microsoft works to keep the Azure environment as efficient as possible, the cloud is the cloud, and there's only so much the host can do. Much also depends on what you're doing with SharePoint.
To improve SharePoint 2013 performance, the first big consideration is page weight. Heavy pages (that is, pages with many large objects, like images) take longer to load, and often require more trips back to the server. Average page weight on the Web overall has increased significantly over the past five years, from just under a megabyte per page to more than twice that. Sure, users expect prettier pages as the Internet evolves, but companies can make pages attractive and populate them with useful content without making them unbearably slow. Employ content management standards such as size limits on images and use caching (where static content is stored somewhere besides the server) to make pages load faster.
Use of folders in libraries is also a no-no. Old-style folder hierarchies work against SharePoint's architecture rather than with it: Libraries are constructed to make nested folders unnecessary, with metadata-driven views serving up the files without endless clicking through folder trees. The inefficiency of folders derives from the fact that folder-nesting foils SQL Server's default indexing scheme, making the actual location of the desired folder an increasingly inefficient operation. Go the extra mile to eliminate nested folders in SharePoint and use libraries as they are designed to be used.
Finally, there's customization. While it's possible to heavily customize SharePoint and fine-tune a carefully crafted user experience (it's all .NET under the hood after all), consider that the purpose of SharePoint is to make that kind of effort unnecessary. There's a lot of overhead in SharePoint's click-through configuration feature and it doesn't go away just because you're coding your site branding from scratch. Customization, more often than not, only adds more overhead, slowing your pages. Best practice: Go as far as you can with out-of-the-box functionality, and customize only what you absolutely can't get otherwise.
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