If I search in contacts, I can type any name but the first two letters appear on the screen and then their is a lag of 10seconds before the rest appear and it begins to list name matches.
I am sure the issue is the iCloud and think it sends the information to iCloud and searches on there rather than on your phone!
In Settings, I only have iCloud selected with contacts turned on. If I turn iCloud off and select it to delete all contacts on the phone (it leaves them on the cloud so you won’t lose them). Then you can add a few contacts on your phone, and when you now search it types instantly etc. When you then click back on settings and turn iCloud sync back on the problem returns.
So why does the phone not just have a setting to let you search for you contacts that are just on your phone?
(I have had this problem for a few years now, so it is not just the current IOS, and I have an iPhone 5)