It's no secret that paladins have a very high burst damage thanks to Divine Smite (and the ability to use it on every melee weapon attack as long as you have spell slots to burn), but this has proven to be quite a problem to me.
My sessions have an average time of 3 hours. I try to give a little of something to everyone every session and make most of the little time we have, so this means I don't have time to throw encounters at them just to drain resources.
I typically only have 1-2 combat encounters per adventuring day, sometimes it gets to 3, it's not like this is a rule or anything, it's just that there's as much encounters as it has bad guys to defeat. I just don't add encounters "because I need to wear them down". I only let my party rest at the end of an adventuring day, not at the end of a session.
Most of the time they only have one fight per adventuring day, and that means the Paladin can go all out with his Divine Smites without caring much for how many spell slots are left, they are lvl 10, so that's a lot of slots to spend. He just takes all the d8s from the table and mows downs whatever is in front of him. Meanwhile, the bard is limited to one spell per turn, and the Ranger and the Rogue can't get anywhere near the paladin's damage.
This leaves the other players frustrated, as they just feel like they're a lot weaker than the Paladin. Giving extra HP to the enemies won't solve the issue, it's not about how fast the monsters die, it's about how much damage the Paladin does.
Eventually, the Paladin player had to leave the table and now another new player wants to play a Paladin, and I don't want to repeat that same situation.
My solution for this was to limit the Paladin to a single divine smite per turn, but I don't know how this will affect the class.
Would imposing this limitation make the Paladin weaker than the others?
Note: since this is somehow a matter of much confusion I will repeat here for emphasis: I do not have the party rest at the end of every session only at the end of an adventuring day. I also do not have one adventuring day per session.
Consider the "gritty realism" variant resting rules.
Other answers have addressed how balanced your change to the paladin might be. However, it seems likely you have a more general problem with players not really being challenged for their limited-use resources in the way that the game expects they will be - for whatever reason. You could address this by changing the resting mechanics.
In chapter 9 of the DMG, the Dungeon Master's Workshop, there are a variety of variant rules for changing the tone and balance of the game. Specifically, on pg267, there is the Gritty Realism variant resting rule:
By changing the resting mechanics to take much longer, a few days of adventuring that might previously have been a cakewalk now become a much more strategic affair where the players must carefully balance the resources they use and possibly even pick and choose their battles carefully.