Just now finally getting a look at 5e, after taking a few years off. Last thing I played was 4e, which I knew inside and out when it came to playing a heavy-damaging Rogue.

So in 5e, it appears that as characters level up, the Fighter (who gains multiple attacks) would do much more damage during combat than the Rogue (who is limited to a single attack).

Am I mistaken? What can I do as a Rogue to deal similar damage? I suppose I could go for Advantage every round to increase my hit rate, but all I can imagine to get that is Hiding behind my allies with a "Naturally Stealthy" Lightfoot Halfling.

I'm feeling kinda stumped here. It seems that although my Sneak Attack damage does rise as I gain levels, I actually become less squishy by gaining ways to reduce the damage I'm taking instead. It feels like I should be much more deadly, maybe with being more squishy as the trade-off.

Opinions? Advice? TIA.

What do you consider "similar damage"? By the time Fighters have 2 attacks, Rogues have 3d6 Sneak Attack; by the time they have 3, Rogues have 6d6. Also, a sword and shield Cavalier Fighter built for tanking is going to be very different from a Battle Master, Samurai or Arcane Archer built for damage.

Keep in mind a Rogue only needs to hit once; two-weapon fighting or the Crossbow Expert feat nearly double your odds of rolling most of your damage dice. The amount of damage dice the Fighter rolls is directly proportional to how many hits they can score on their attack rolls.

Also, if you can attack an enemy with your reaction when it's not your turn, you can even use Sneak Attack a second time. There's a couple of ways your friends can help make that happen: Dissonant Whispers, Haste or a Battle Master's Commander's Strike. On the other hand, a Fighter with 3 attacks only deals 1/3 of their Attack action damage with their opportunity attacks.

If you can surprise your enemies (which implies none of your party members failing their Stealth checks), Assassin Rogues can deal big damage on the first round of combat.

Wait, I can use Sneak Attack a second time in the same round? Can I have a rules citation for that? I didn't see anything about that, but I'd like to be prepared if I try to use it and someone tries to tell me I can't do it.

The rules say you can use it once per turn, not once per round or on your turn only.

See Sage Advice Compendium for more details:

Can a rogue use Sneak Attack more than once per round?

The Sneak Attack description specifies that you can use the feature once per turn, but it’s not limited to your turn. The feature also doesn’t limit the number of times you can use it in a round.

You sometimes get a chance to use Sneak Attack on someone else’s turn. The most common way for this to happen is when a foe provokes an opportunity attack from you. If the requirements for Sneak Attack are met, your opportunity attack can benefit from that feature. Similarly, a fighter could use Commander’s Strike to grant you an attack on the fighter’s turn, and if the attack qualifies, it can use Sneak Attack. Both of those options rely on the use of your reaction, so you could do only one of them in a round.

Because you get only one reaction per round, you’re unlikely to use Sneak Attack more than twice in a round: once with your action and once with your reaction.

 That's wonderfully tasty, thank you! Shame they didn't just incorporate that into the regular rules on here after the SAC was released. 

Strictly speaking Sage Advice Compendium isn't errata, but do I wish it were accessible on DDB somehow.

I also forgot to mention another way to get that second Sneak Attack: take the Sentinel feat and stay within 5 feet of an enemy. If it attacks someone else, you get a chance at Sneak Attack; if they attack you, you can use Uncanny Dodge. This is still risky though since Uncanny Dodge only works against one attack, and flying enemies will make this difficult too. Haste is definitely the best method if you can get someone to cast it on you (or you cast it on yourself as a 13+ Arcane Trickster.)

Quote from InquisitiveCoder >>

two-weapon fighting or the Crossbow Expert feat nearly double your odds of rolling most of your damage dice.

How do you figure?  It seems to me this is only true if the odds are very low to start (i.e., a combination of high AC enemy and low attack bonus).  To be precise, the chance of landing at least once goes up by a factor of (2 minus the chance of a hit on a single attack).  For example, a Rogue with a +4 DEX mod, proficiency in their weapon, attacking without advantage against an enemy with an AC of 17 has a 50% chance of getting the sneak attack in (assuming an ally is next to the target) if they attack once, and a 75% chance if they attack twice.  In this same scenario but attacking with advantage, they have the 75% with one attack and a bit under 94% with two (assuming advantage applies to both attacks). 

But the fundamental DPR point holds: take a level 8 Rogue with a hand crossbow and crossbow expertise vs. a level 8 Fighter with a greatsword in the scenario above (assuming matched attack and damage modifiers).  With no advantage, the rogue attacking twice has 16 expected damage; the fighter attacking twice only 9.  With advantage the rogue's expected damage is over 21, the fighter's is 13.5.  Even if the rogue only attacks once, their expected damage is still a bit higher than the fighter's (10.75 without advantage, over 16 with), and they get a cunning action on top of it (which, focused only on offensive value, they might be able to use to get advantage next round).