Hello Airsculpture,
Welcome to Apple Support Communities. This may be a connection issue, or could be that the Apple TV may need a few things to happen.
First, let's restart the Apple TV.
Restart Apple TV - Apple Support
"Do any of the following:
- Press and hold both the Back (second-generation Siri Remote) or (first-generation Siri Remote) and TV buttons on the Siri Remote until the Apple TV status light blinks rapidly.
- Disconnect Apple TV from the power outlet, wait five seconds, then reconnect it.
- Open Settings on Apple TV, go to System and select Restart."
Restarting will give your device a fresh start, and could resolve the issue. Also, have you tested this on another Wi-Fi network, or another band on same network?
- Go to Settings
- Go to network
- Tap Wi-Fi at the top
- Choose a different band if applicable.
For example, if you have a 2.4 and a 5ghz band, choose the one you aren't currently using and see if the connection is better, and stops the lag.
If that doesn't help, please let us know and we'll take a look at other options.
Cheers!