Resource Management during the Adventuring Day
As a class, the Paladin isn't overpowered. The Paladin does have a nice mix of weapon and spell skills to offer any party.
On page 84 of the DMG the adventuring day around which the game is balanced calls for 6-8 encounters of medium to hard difficulty, and 2 short rests. If your DM isn't running that much challenge, then the Paladin's resource management problem isn't being put to the test as the game balance intended it to.
A Paladin can do large bursts of damage, for sure. What he can't do is burst damage all day, every round, for that many encounters.
Melee Attacks Available
You mention that the Paladin is now 5th level. This is a threshold in melee damage increase thanks to having gained the second attack per melee round. The initial increase in apparent power should level off a bit as you increase levels and the DM increases the CR of the monsters you face.
He has almost died at times
If he's playing "high risk - high reward" style, over time the odds will catch up with him. It appears that the shield spell that goes with his armor is giving more chances to not get hit. He is thus able to play a higher risk combat style than he might otherwise. If it allows him to do two handed weapon attacks, and still benefit from a shield through magic, that's more powerful than a paladin who has to use a shield and a one handed weapon to handle the rigors of melee combat.
That combined with the great weapon master feat might be what's causing this perception of great power. It might also be the pace of your adventuring day.
Per Long Rest, at 5th level, he has 4 spell slots of 1st level spells and 2 of second.
If he doesn't have to concern himself with casting any other spells (this has to do with your party make up and who the other spell casters are) then the paladin can use divine smite six times per long rest. That's a nice pile of bursty damage during some fights.
Is that "overpowered?" If you only have a few encounters per day, it can look that way. This is an adventure design challenge for the DM to address, as you both suspect.
Encounter design is explained on pages 81 - 85 of the DMG. The proposed XP budget for encounters and the adventuring day is built around a party of 4 characters, and the challenge ratings that accompany that XP budget. (For a party of 5th level adventurers, that's 3500 XP per character per adventuring day, 14,000 XP worth of monsters). If the DM reviews that and compares it to the challenge he's been offering you, he may choose to ramp up the difficulty.
A character can only have one long rest in a 24 hour period [PHB, p186]. This serves to limit the power of abilities and spells.
FWIW: in our party, the Paladin is often either the hero or the goat. He who lives by the burst, dies by the burst. He has a +1 magic axe (just found, has not used in battle yet) and some +1 bit of dwarf light plate. He does not overpower the rest of the party but he can hold his own.
In a tweet Jeremy Crawford (lead rules designer) advised:
Q: Is divine smite supposed to be limited to once per turn? Can I use it three times with extra attack and DW?
A: The limit on Divine Smite is the number of spell slots you have available to spend on it.