Can I use a brighter bulb in a Harbor Breeze Caratuk fixture?
Pretty low wattage for a lamp socket. I have never seen one listing that low. This would mean LED only. But, if this is the case, you could potentially choose a higher temperature range to get "brighter" light. If you have warm white, change to cool white. Try to find a bulb in the 5000k-6000k range.

I looked up the fan on https://www.lowes.com/pd/Harbor-Breeze-Caratuk-River-52-in-Bronze-Indoor-Downrod-Or-Close-Mount-Ceiling-Fan-with-Light-Kit/50354992 , and it states a max wattage of 60. So you could get a 100 watt equivalent LED and install that. Incandescent, CFL and LED are all different. At 60 watts, you are limited to a 60 watt incandescent, a 60 watt CFL is equivalent to a 225-276 watt incandescent, and a 60-watt LED is roughly equivalent to a 276-450 watt incandescent. Those are all estimates.