Recently on several Sharepoint Online sites which belong to different tenants we encountered with similar problem with slowness. Slowness didn’t appear immediately, i.e. from beginning sites worked fast. But after some time response time grew to 6-14 seconds, which made sites almost unusable. First of all we checked SPRequestDuration response header in browser developer tools which shows server response time (see Diagnosing performance issues with SharePoint Online). It showed that slowness is not caused by client side custom components but instead comes from server side because SPRequestDuration had value from 6000 to 14000 milliseconds. All sub sites used OTB seattle.master master page without customizations. After deeper investigation we found that the problem is caused by structural navigation used for left navigation (for top navigation managed metadata navigation was used). I.e. with time number of sub sites in hierarchy grew up and it made structural left navigation work much slower.
We opened ticket in MS for that, but as workaround for this issue you may change left navigation from structural to managed metadata. After we did it on all sub sites (as in Sharepoint single term set with navigation terms can be used on single site, we set managed metadata left navigation on the root site and then inherited left navigation from the parent sites on all sub sites. This approach also has own problems, because in OTB master page only 2 levels of hierarchy are shown. Better approach will is to create own navigation term set for each sub site, but there you also need to think how to keep navigation terms synced with site structure), server response time was reduced to 1 second. In one of the future posts I will describe how you may keep in sync sites structure and managed metadata navigation terms hierarchy.