I haven't tried this on every browser, but what works for me is to use the keyboard shortcut for select all (CMD+A on Mac) while the notebook is selected. That should select all the cells in the notebook which you can then delete. If you use the browser's edit menu, select all, rather than the keyboard shortcut it will not work (because we can't override the menu behavior like we can the keyboard shortcut).
Here's another trick if you just want to delete a block of cells. Use CMD+4 to insert a section cell before and after the block you want to remove. Now you can just select the outer section bracket which contains all of the cells delete everything. You can remove the second section by selecting the inner bracket that contains only the section cell.