The confusion here comes from not staying consistently in one or the other of the two frames of reference present.
In the earth's frame of reference, you are riding on a rotating merry-go-round.
You are accelerating towards the centre of rotation of the merry-go-round; the direction of your velocity vector is constantly changing.
By Newton's First Law there must be an unbalanced force acting on you in the direction of your acceleration. The immediate source of this force is the friction between your body and the seat. Ultimately, the force comes from the radial tension force in the structure of the merry-go-round.
Importantly, there are any number of instruments that can measure the size of this inward tension force. We could trim the whole merry-go-round down to a central pivot, a rope with a spring scale as a link in the rope, and the passenger.
Let's change to the frame of reference of the merry-go-round. This means that the merry-go-round is at rest, and anyone sitting on the merry-go-round is also at rest, and is not accelerating. It's the rest of the amusement park that is rotating!
So, again applying Newton's First Law, there must be no net external force acting on the at-rest passenger.
Unfortunately, those pesky instruments are still indicating that there is an inward force acting on the passenger. So, to save Newton's First law, we create an outward centrifugal force with a size to just balance the real inward force.
Newton is safe, at the cost of creating fictitious forces...