High CPU temperatures after replacing thermal paste?

I own a Dell XPS 15 (L521x) laptop.
I replaced the thermal paste because my idle temps were around 55°C (with a cooling pad and average room temperature).

Under stress like games such as running Minecraft (and running my private server), or other games on Steam, my CPU's high would be around 85-85°C at most. Fan would run normally.
Now I know laptops aren't meant for gaming (mainly because of heat, especially high end processors create A LOT of heat in a small confined area)

Anyway, I just got Coollaboratory Liquid Ultra thermal compound. I applied it myself, taking all precautions and did every procedure correctly…
-Blew compressed air everywhere in the laptop before taking off the heat sink
-Took off heat sink
-Used alcohol and a Q-tip to rub off old thermal paste on CPU/GPU and heat sink (as well as using a lint-free microfiber cloth)
-Gently applied a very small dot of the stuff (I have a small CPU/GPU) and used a Q-tip to spread it out evenly, of course there are microscopic bumps but it can't be 100% perfect.
-Waited a minute for it to dry(?) making sure all fans and windows were closed so dust doesn't move around
-Reattached heat sink in the correct screw number order (1-6)
-Turned on computer, left it idling for around an hour (maintained mainly 45°C). When I played the same games, the temps would peak up to 93°C!

Did I do something wrong? Or is it normal after applying new paste? Wait/break-in period?

"used a Q-tip" - you probably trapped a fiber.
repeat your procedure, with some changes:

-Blew compressed air everywhere in the laptop before taking off the heat sink
-Took off heat sink
-Used alcohol and a Q-tip to rub off old thermal paste on CPU/GPU and heat sink (as well as using a lint-free microfiber cloth)
CHECK FOR FIBERS
-Gently applied a very small dot of the stuff (I have a small CPU/GPU) " not too small!
AND DON't SPREAD IT. The contact with the cpu will make it spread out.

Reattach heat sink in the correct screw number order (1-6)

now turn on computer, left it idling for around an hour