Some notes:
You only need to beat up to chapter 4 before you unlock the DLC island and can grind to your heart's content. Of these, chapter 2 is the only one that is especially difficult
Funnel as much experience to the Avatar as you can. A +Speed or Defense asset is probably the most useful, and Luck seems to be the most popular flaw.
You can get a lot of experience to the Avatar on the prologue by doing the "water trick". The gist of it is that after clearing the first wave of enemies, move everyone into the river (paired up), and use the Avatars magic to kill all the enemies.
Consider soft resetting if you don't get speed or defense level ups. This can be frustrating so I don't necessarily recommend it, but I would definitely reset if you get any bad levels up.
Frederick and Avatar should be the only ones seeing a lot of combat. Yes, you can use the other people to finish off a weakened enemy or take a hit, but make sure they won't be exposed on the enemy phase. Fred and a leveled Avatar are the only you can safely send into the range of multiple enemies and still expect to come out alive.
Using the weapon triangle matters. Though you don't have a lot of inventory to work with, you can trade Avatar's bronze sword to Frederick when he will be under attack by axe users.
Terrain like forts and forests really help a lot. Even +1 or 2 defense is a big deal when everyone's stats are so low.
Consider soft resetting until you get some useful items on the event tiles. Something like a Finn's Lance or Roy's Blade in the hands of Frederick can really shift things in your favor. Any of the confects are great too.
Chapter 2 is the big hurdle you'll need to get across. There's no guaranteed way to do it, as a lot of it comes down to luck. Basically you wan't to move Frederick and Avatar paired up (whichever one has higher defense should lead) into the range of as many enemies as you can, and move everyone else to safe squares. You just have to hope that you can kill a lot of them, because if Frederick/Avatar are surrounded such that enemies can't attack, they will start moving toward your weaker units.
You can through Ch 2 with less frustration if you decide to use a unit as a sacrifice. Stahl or Sully probably works best. Basically you want to move them away from your main group of weak units, into the range of enemies. This can buy you a couple turns because enemies will move back to deal with them instead of moving towards your group. It will also soften a few enemies up a little, and will divert a few hits that otherwise Avatar or Frederick would have had to take. But since the loss is permanent its not something I will totally recommend. If you are playing on Casual mode, this is actually the best strategy, as it doesn't matter who dies as long as it's not Frederick, Avatar or Chrom.
Thankfully, clearing the second group of enemies is not nearly as difficult, Just lure them towards you, and move back into one of the forts, leaving your weak people way far back. Now the hard part is over.
Chapter 3 is a cakewalk in comparison. Only the very beginning is tricky. After that, just use the tight corridors to tank enemies one at a time while Lissa stands behind them and heals. I'm not even going to mention Chapter 4, as your Avatar should be leveled enough to not really worry about much. Then you're home free to the DLC.