Centripetal force is NEVER directed outwards. It is the force that makes the object travel in a circle.
It is always directed towards the centre (which is basically what the word 'centripetal' means), and for uniform circular motion that is the only net force.
You should be calling the direction radial, NOT horizontal. In a horizontal circle, 'horizontal' could mean anything.
For non-uniform motion, there is also a tangential component. The vector sum of the radial and tangential components won't point at the centre, but it will never point outwards.
Other than that, I can't offer any more help without a specific question from you.