How do I disable normal chat?

local Chat = game:GetService("Chat")
local function setUpChatWindow()
    return { BubbleChatEnabled = true }
end
Chat:RegisterChatCallback(Enum.ChatCallbackType.OnCreatingChatWindow, setUpChatWindow)

I know this is how to add bubble chat, but how would I remove normal chat

You should be able to do this the exact same way (via RegisterChatCallback). It is referred to as ClassicChat, and what you need to disable is ClassicChatEnabled.

local Chat = game:GetService("Chat")
local function setUpChatWindow()
    return { BubbleChatEnabled = true ; ClassicChatEnabled = false }
end
Chat:RegisterChatCallback(Enum.ChatCallbackType.OnCreatingChatWindow, setUpChatWindow)

This should do the trick. Note that if you disable both BubbleChat and ClassicChat - the text bar will not appear at all.

So first

Insert Object >> LocalScript >> StarterGui Next <<

<< Paste This Script In >>

game.StarterGui:SetCoreGuiEnabled(Enum.CoreGuiType.Chat,false)