Change order of iMessage Apps?
Rearranging the apps in the grid doesn't affect the order that they show up when you're swiping side to side between iMessage apps. My guess is that Apple expected people to largely use the app draw to select the iMessage apps they want to use, with the horizontal scrolling view functioning more like the app switcher for the phone, with the most recently used app appearing first.

So the horizontal order is based in the order in which you've opened the iMessage apps from the app draw - with the most recent at the left and oldest at the right. To get the order you want in the horizontal scrolling view, first open the app draw. Select the one you want at the right hand side. Then reopen the app draw and select the one you want to the left of that, and repeat, ending with the app you want at the left hand side of the iMessage apps.  

So to get the below order mapped over to the horizontal view I'd open the app draw and select '#images.' Then reopen the app draw and select 'grammarsnob,' reopen and select 'giphy,' reopen and select 'Alto's Adventure.'  

  

It's not very intuitive, but I guess they're going for Snapchat levels of UI confusion.