I think it is also very related to the networks, as we are now using 4G, the speed become much quicker than before, and share a video is also much fluently. Now i am using the Samsung Galaxy Note 3, it is fluently for me to upload a video or start a chat with friends. Other aspects that you may need to check are:
1. whether there is a problem with the iphone 5 setting.
2. is it because your video form?
3. Close and restart your iphone mobile.