A VFR pilot departing a Class D airport reported a NMAC with crossing traffic during initial climb out. They did not receive a traffic advisory from the Tower.

Narrative

I was given my clearance to take off from Runway XX after informing Tower that I would be departing to the northwest. I had climbed straight out to about 700ft. and then turned about 15 degrees right while still climbing out; when an airplane appeared in front of me from my left to right. I believe it was a white and blue light single. I had to make a moderately aggressive turn to the left; otherwise I'm not sure a collision would have been avoided. That airplane was at my altitude and only 100 to 150 feet in front of me. I had not been warned of this potential conflict by the Tower and no mention of it was made by ATC afterwards. As I left the Class Delta; the controller advised frequency change approved. I did not mention it either; although shortly afterward I wondered why I didn't speak up immediately. I think I simply dealt with it and that was that; and had it occurred at a non-towered airport I would have just chalked it up to busy airspace. But this should not have happened within a Delta.