Apple carplay not working on new iphone 13 on IOS16.5.1 (a)

If CarPlay doesn't activate automatically, look for the CarPlay logo on your car's display and select it. Restart your iPhone and your car. Make sure that Siri is on. If your iPhone isn't detected by CarPlay, make sure that CarPlay isn't restricted.

If you need help with CarPlay - Apple Support (IN)

Apple has withdrawn the iOS/iPadOS 16.5.1(a) and macOS 13.4.1(a) security measure.

See -> About the security content of Rapid Security Responses for iOS 16.5.1 and iPadOS 16.5.1

About the security content of Rapid Security Responses for iOS 16.5.1 and iPadOS 16.5.1 – Apple Support (UK)

• Apple is aware of an issue where this Rapid Security Response might prevent some websites from displaying properly. Rapid Security Response iOS 16.5.1 (b) and iPadOS 16.5.1 (b) will be available soon to address this issue.

• You can choose to remove Rapid Security Response (a): In Settings > About > iOS Version, tap "Remove Security Response." Then tap Remove to confirm.