General aviation pilot reported a near miss while under ATC control. The pilot misunderstood ATC instructions to provide separation.

Narrative

I was on a visual to VNY handed off by Burbank tower to Van Nuys Tower. VNY advised me to enter the left downwind for [Runway] 16L on a 45. I was headed exactly as asked when the tower then told me to turn north to go up the 405 [freeway]; 'if possible'. I was already headed to do exactly that; and not sure why he gave that further instruction. It was confusing; and made me think I was already passing the 405. I started to turn to what look like the 405 a large thorough fair. As I rolled out; I realized the 405 was off my right wing and I needed to go further west. Right at that moment; the controller told me to make an 180 immediately. He did not give me a direction to turn; and I didn't understand why. I repeated his instruction and made an immediate turn as requested; thinking there was an issue with traffic. I went to the right as not to fly right at VNY. The tower then instructed me to fly to the 101 [freeway]. He queried again and said 'I told you to fly south'. I let him know that is exactly what I was doing. It turns out that this maneuver caused me to come within close proximity to an airliner going to Burbank. In hindsight; it did not make sense to have me do an 180; all the Tower need to do was have me come west; and continue and I would not have come close to the airliner; and I would have been clear of his flight path; and we would have been deconflicted. For my part; I will try to clarify the request; and ensure I am seeing the exact item he is referencing in 'ground terms.