h3 instead of Nc3 is better. There's also a case for waiting with castling so that you can play h3 and g4 (especially when your opponent has castled kingside).
Kingside knight pinned against queen
Here's a more general video on this motive, possibly not in this line in particular and actual move order is important, but these general ideas are good: https://www.youtube.com/watch?v=l_zWJHYWWJs
You do want to play h3 to stop the pin in positions where you're trying to go c3, d4, though. Also mostly in the Guico Pianissimo sort of positions you just shouldn't castle so early as the pin can actually be bad. You don't always want to push the g-pawn as in the video if you're castled as the the opponent might sac a piece for two pawns in front of your King and sometimes the attack is good. Those things said, that's a good video series, IMO. If I remember right there's more good stuff regarding that pin and the ruined Kingside you get from it later in that series.

Your opening play is weak. The so-called "Italian Four Knights" is equal at best for White.
The rule of thumb in double king-pawn openings is that of the 3 moves, Bc4, Nc3, and Nf3, no more than two of the three moves should be played early. Otherwise, Black gets complete equality and possibly more very easily due to multiple issues. The "Fork Trick" and the complete lack of pawn breaks for White!
Take a look at the following:
The Vienna: 1.e4 e5 2.Nc3 Nf6 3.Bc4. This allows the Fork Trick, 3...Nxe4, known as the Frankenstein/Dracula Variation, but White shouldn't take on e4, and instead play 4.Qh5. Note that with Nf3 played, this wouldn't be possible, nor would any f4-break.
The Italian: 1.e4 e5 2.Nf3 Nc6 3.Bc4. Here, Nc3 should not be played because White needs to play c3 to break "soundly" with d4. The Knight on f3 blocks all f4 breaks.
The Four Knights Game: 1.e4 e5 2.Nf3 Nc6 3.Nc3 Nf6. Here, Bc4 is not recommended again because of the fork trick. White should either play 4.Bb5 (known as the Spanish Four Knights), 4.d4 (known as the Scotch Four Knights), or 4.g3 (known as the Glek System).
In each of those openings, two of the three moves are played, but never all 3, and in the Ruy Lopez, only one of them is played.
However, 4.O-O isn't best. 4.c3! is White's strongest move (again, Nf3 and Bc4 are already played, Nc3 is now a no-no).

Avoid this opening, it's called the guico piannisimo, it's a playable opening and it's completely fine (not bad at all) but its very boring compared to what's available.
If you like to put your bishop on c4, then play the Greco variation instead. In that opening, instead of Nc3, you play c3.
And for something more double edged, the Evan's gambit is also a fun opening to go down.
To stop the pin play h3 early.
I recommend the pdf "Exeter Chess Club: The Italian Game for beginners".
Don't let the "for beginners" fool you, it's a pretty useful guide and it goes through a bunch of games to show the general ideas. It also shows examples of what not to do. It takes a while to digest it all and play through the games, but once you read it you'll know how to take the most out of this opening.
My opponent often uses his bishop to pin my f3 or f6 knight against my queen after I've castled kingside, and I don't know of a good way to deal with it. If I chase it away with my pawns, I expose my king. If I move the queen away, he can take my knight and force me to open my pawn defenses to recapture. If my bishop isn't already on the other side of my pawn chain, I can put him between my knight and queen, but then he'll get blocked in if I push my e-pawn. Surely there's a better way to handle this?