Increase in measurement of magnetic field lines due to increase in current

Magnetic field is AROUND the current carrying conductor, not inside it (well technically it is, but it radiates outward). The magnetic field around it is traveling away from the conductor at the speed of light, thus reducing in strength. This is why strength of magnetic field decreases with distance. Magnetic field lines are fictional and are just used to simplify imagining the strength of magnetic field (more lines in small space mean bigger field strength).

When the current increases, the strength of magnetic field also increases, thus if you want to use your teachers analogy, the number of magnetic field lines in a space around the conductor increases.

A stronger field is sometimes represented by more magnetic field lines passing through any given area. So with more current in the conductor the lines will get closer together! But as Ma Drung says, the lines are fictional, insofar as one can draw as many or as few as one likes. In cases like a current-carrying coil, if one decides (arbitrarily) on a number of lines passing through a given area near the centre of the coil, the lines do get further apart as we follow the lines as they curl round the coil – where the field is weaker. [Unfortunately we can't do this for the field lines round a long straight wire, as the distance apart of any two lines stays the same as they curl in circles around the wire. If we want to represent the strength of the wire's field by how close together the lines are, we have to do this 'artificially' by choosing the spacing of the lines ourselves.]