Toilets, Sinks, Showers, Tubs and Disposals - Toilet bowl filling quickly and not draining timely

I flush the toilet and the water fills the bowl very quickly, yet it doesn't go down the pipes to drain quick enough. So the water level rises very quickly at which point, I turn off the valve under the toilet to avoid an overflow. The water level in the bowl then eventually lowers to a seemingly normal level.

I initially thought it must be clogged--with what, I don't know. I hadn't put anything in the toilet that doesn't belong there (and I live alone with no children). I tried two different plungers and also used a toilet auger...but to no avail! :(

I tried researching online and decided to try pouring a bucket of water into the bowl to see if it would flush...and the water in the bowl did empty. So does that mean that it is not clogged and it is something else?!

What might be going on here? Any suggestions would be GREATLY appreciated. I have already spent a lot of time on the internet trying to find out what is going on...but haven't come up with a solution yet. Maybe I will need to call a plumber?

How much water was in the bucket & how much water does the tank hold? The flapper at the bottom of the tank should stop the flow into the bowl in a few seconds. If it continues to the point where you have to shut the valve, something isn't adjusted correctly.