Most codes require at least 15 inches (measured from the center of the toilet) from any side wall or obstruction and not closer than 30 inches center to center to any other sanitary fixture. (The NKBA actually recommends 32 inches.) There should be at least 24 inches of clear space in front of a toilet or bidet.
Sourced from This Old House, emphasis mine.
You’re going to need to pull the bowl off the carrier/wall. Based on the fact the leak appears to be towards the top of the bowl, I’d check the filler tube/inlet. Make sure it’s cut to the correct length and make sure the gasket is greased so it goes in straight and smooth. I seriously doubt it’s the carrier.