In the size bowl and it is most likely 1-2 gallons..just guessing here....
But yes, it sounds like a water quality issue by just your water change schedule and that is an easy fix
To treat the tail problem now and to help encourage re-growth and prevent a secondary infection-I would start by making 100% daily water changes for the next 7 days then........regular scheduled water changes of-
Unfiltered, 1-2g or even 3g....Regular water changes of 50% 1-2 times a week and 1-100% water change a week-totaling 2-3 a week water changes to keep the water quality high enough to help prevent health and fin issues.
Always use the dechlorinator with any new water added to the tank/fish if you are on city water supply and keep the water temp within a couple of degree from new and old water to prevent any water temp related problems.
The plants-if they are plastic you may want to remove them due to the roughness-they can be part of the problem by snagging and ripping his delicate fins....... silk or live plants are often better to use with this species of fish.
Water temp can also be an issue and unless you live in a warm climate area you may need to invest in a heater and thermometer to keep a stable water temp in the 78-80F area
If the tail starts to get worse or behavior changes you may need to start a treatment-I use a natural treatment with Aquarium salt 1tsp/gal along with 100% daily water changes for 10 days.
Would love to see pic of your Betta and your set-up....
And welcome to the wonderful world of Betta keeping.....