Why is my iPhone showing LTE instead of Wi-Fi?

Are you running a Personal Hotspot to share your connection with some other device(s)?

If you turn on Personal Hotspot (Settings -> Personal Hotspot), then the phone won't auto-switch between cellular and WiFi connections since that would interrupt connected clients.

Have you checked to see if you are connected to a Wifi network? If your phone is showing LTE, it is connected to cellular. And your phone will automatically choose the best connection to ensure you can be reached or connect to the internet. Go to Settings > Wifi and see if you are or can connect to Wifi.