Can someone please correct me?
There is nothing to correct. You are not wrong.
Say your charge is negative.
If you put a positive charge exactly on the negative charge with distance $r=0$ between them, then yes, the force is quite large. It is more difficult to bring them appart than if there was a larger distance. So, there might happen some kind of merging now (which is for another topic), since the force is so great.
Surely, you cannot have exactly zero distance, but your theoretical thinking is correct. Consider a field drawing. Where field lines are closer, the field is stronger. When they meet (which will only in theory be exactly at the charge location) electric field would in theory be infinit.
From the elaborated question in the comments:
Why this charge is not affected my its own field which in this case should be infinity since $r=0$?
Consider a charge like a planet. The planet flows around in space and just is there. It doesn't pull in itself; it's gravity doesn't make itself accelerate.
Now, if a large asteroide is near, then the planet starts to attract this astroide. It pulls in it. This will move the planet too. From Newton's 3rd law, the force it pulls with is also acting on itself, pulling itself forward with exactly the same force. But in the opposite direction.
Now, think what would happen if it pulled in itself with its gravitational force. It will pull itself in "some" direction. But at the same time, from Newton's 3rd law, the exact same magnitude of force will pull it forward in the opposite direction. There will be no net force.
Similarly, when I pull up in my own hair, I will not fly. The exact same force, with which I am pulling, is also pushing me downwards.
Similarly for any force from any "object". Similarly for charges. Nothing can exert a net force on itself.