I'm new to the forum do forgive me if this has been raised before.

What do you do if a PC is killed during the course of a campaign and surviving characters are say, level 4+

Introducing a new 1st level character is going to be very hazardous to that character and they are going to lag behind everyone else for a long time.

What options are there? 

Usually the course of action I would take is to work with the characterless player to create a character that is the same level as the rest of the party, and create a coherent way for them to be added to the story and join the party.

There are three ways of handling this that I know of. They are as follows:

1. The player makes up a level 1 character. This is the "penalty" for dying (as if losing your character isn't bad enough). Defenders of this method argue that the XP curve goes up so steeply that the level 1 will catch up really fast. For example, you get to level 4 with 2700 XP but level 5 takes another 3800 total. By the time the 4th level party hits level 5, the level 1 character will have gained 2, 3, an 4, and be partway to 5 and only 1 level below. I am not a fan of this method, but it is the "old school" way of doing it.
2. The player makes up a character 1 level below the rest of the party. (I.e., a level 3 character joins a level 4 party.) This is basically just a less harsh version of the first option, but makes it much less likely for the character to die again before leveling (as a level 1 character facing things like trolls might do). Usually the level-1 character is given a few basic magic items, nothing impressive, but enough to make him or her roughly equal to where the party was 1 level ago. 
3. The player makes up a character the same level as the party and gets a set of magic items and other treasure equivalent to the rest of the party. This is how I would do it (no deaths yet in my campaign to level 3, knock on wood). IMO, the death of the character is enough of a punishment. I don't need to make the player suffer more by being behind everyone else, and worse, make the player's character the most likely one on the board to die again.

Personally I see no reason to go with the first 2 options but, those are the 3 methods I have seen used (or used myself back in the day -- old school it was make up a new level 1).

I don’t give XP equally to all PCs. For events and encounters it’s pretty much equal, but boss kills and strong RP earn a little extra. So it is possible for PCs in campaigns I DM to not all be the same level. The rule I have is that new/replacement PCs start at the minimum XP to be the same as the lowest level PC in the party.

So if the everyone in the party has between 6,000 and 7,000 XP, then the lowest level PC in the party is 4th level. So a new PC would start at the minimum XP required to be 4th level, or 2,700 XP.

When adding a new character to an on-going campaign whether due to a character death or simply gaining a new player, I will generally start them at 1 level lower than the main party AND give them a specific link in the form of a goal or task they are seeking to accomplish.  This not only provides them with a reason to join or work with the existing party, it gives me a milestone unique to that character that I can then use to grant them the extra level.  That way after a few sessions, they should reach the same level as prior party members and have formed a connection with the party that explains them joining permanently.  

Some of the choices suggested here are... a bit harsh. In the group I play in, the killed character is replaced by another character of the same level, with no magic items/special equipment or extra money. Other players usually loot the dead body of magic items.

Assuming the replacement character could use it, we always gave the person's new character his old character's magic items. It may not have been "in character" but it felt fair to us to do it this way.

I vote for No. 3 from biowizard’s first post. Bring them in at the same level, with level appropriate gear. There is no real reason to punish them with a lower level when a character dies. 
The only reason I’d ding someone a level is if they repeatedly decided to make new characters, not dying, just making a new character. I’d even allow that once or twice, because sometimes you make something and it’s not as fun as you thought, so I make allowances. But that has its limits. 
But for death, just bring them in at the party level. It’s so much easier. 

My thinking is definitely same level or level minus one. It seems that the death of a beloved character is probably pain enough. Thanks for the suggestions, people. 

Why wasn't the character simply resurrected? At 4th lvl, the character should have enough resources to afford a Raise Dead service from a temple. If the PC had a good reputation with someone with wealth or a religious affiliation, all the better.

That'll depend on the world. In my world, most temples do not have a cleric who can raise dead, because "clerics" are adventurers and not just priests at the temple. The high priest of the main temple in a major city (e.g., Rome, Carthage, Alexandria) can probably do it... but in town or small city, probably not.

I say you shouldn't try to add characters just to keep them safe. If one of your characters dies, you should probably have them come back at the lowest amount of everything that the other players have. Like for instance, if one of the players had 6 magic items and the other 2, the new player would have 2 magic items. I feel it isn't unfair to the player who is coming back that way, and having too much would be unfair for the other players, because they had to work to level up and they probably want their characters to be a bit better than the new ones coming in.

Why does one character have 6 magic items and the other 2, though? Presumably the party divided the loot up that way. Maybe the 2 items are a Deck of Many Things and a Staff of the Magi, and the 6 are just +1 weapons and armor. Maybe the party gave all the items it did to the one guy because nothing "special" has come for his class so he is getting all the cast-offs... but the guy with 2 has 2 items just for his class, like a Holy Avenger for a Paladin. Just THAT item might equal all the rest of the magic items in the entire party combined, in some cases.

I wouldn't count up magic items per character. I leave it up to the party to decide who gets what. I'd look at what the party has, in total, and then give the equivalent of 1/N worth of magic items to the new character (where N is the party size not including the dead or new guy). So if a party of 5 had 20 magic items, that's an average of 4 items per. New guy gets 4, of roughly the same power level. For instance, if there are a bunch of potions and scrolls then he gets a random potion and random scroll... If there are a bunch of +1 weapons, he gets a +1 version of his proficient weapon... and so forth.

This all seems really harsh or over complicated. Just make a new character of the same level, give them appropriate magical items, and introduce them into the story. That's generally all you have to do.