General aviation student pilot reported a near miss with another aircraft in the traffic pattern. The pilot followed ATC instructions to separate from the traffic then continued the VFR flight.

Narrative

On the route to RNM from ZZZ I had experienced more turbulence than I have ever experienced in my career of flying in route over Barona Casino landmark. It didn't seem enough to cause a disturbance but it definitely was concerning as I was on my fourth solo flight and didn't have my instructor to reassure me this was normal (on the ground; he did in fact reassure me this is totally normal and well within safe limits assuming maneuvering speed etc). RNM was also especially busy that day; so I was trying to be extra mindful. Upon landing R27; and taxing to runup to double check my mags; I requested a clearance absentmindedly without realizing that three planes were taxi's up to the hold short line so after my take off clearance was granted I had to inform Tower I didn't have access to the runway and to let some of the other planes depart first. This mistake made further distracted me. After waiting some time; and when the way was clear I taxi'd to the hold short line and requested clearance again. I was worried about wasting the Towers time when it was so busy so I tried to expedite my instructions when I got onto the runway. I also forgot to close my door ahead of making the request so was fumbling to do my runway items (fuel pumps; mixture; lights) while also trying to close my door; then as soon as that was sorted I was full throttle and overly focused on checking my airspeed was alive; my gauges were green; and rotating and climbing at the appropriate rate. My take off was without incident but the issue was I was not tracking or checking for the traffic that had departed just before me. They must have been in the upwind at the time of my take off. I had remembered that there was some noise abatement preference to start your turn for the downwind a bit early so as not to fly over certain homes/structures but in my rush I instead started my left crosswind a bit early. While I was scanning for traffic I did not see the aircraft ahead of me in the pattern that must have been on its crosswind some distance upwind. When I carried my crosswind through and came into the downwind was when I realized the aircraft was abreast my wing and maybe 100-200 feet off of my right side. That shocked me a bit and was also when Tower called me and told me to follow the aircraft in the downwind. I informed Tower that I could not do that as they were abreast and asked what they would prefer I do. They told me to descend and I began descending; reading back the instruction while also reminding them I was a departure to the southeast. Tower then said; stop descending; and remain eastbound. I stopped descending; wasn't sure if I was supposed to ascend or maintain altitude so I began ascending again and read back the stop descent with a follow up that I was in fact ascending. They affirmed my instruction and I just continued my course checking for traffic until they fell away behind me. (I believe they were given right 360 instruction but not sure).At the time; I wasn't sure exactly what had happened. I assumed the other aircraft was arriving into the pattern and we failed proper seperation. But looking at an ADS-B radar log of the time and event proved that I had simply started my crosswind far too early and effectively cut the corner to end up abreast the other aircraft.This is was purely and completely pilot error. I was flustered; saturated with tasks; and was well behind the plane. Looking back on everything I should have taken more time to get completely ready for takeoff. I should never request takeoff without being fully capable of looking for traffic in the pattern; and noting anyone departing ahead of me. Furthermore I need to be extremely careful when it pertain to early cross winds (something that should almost never be done) and remind myself if I start to turn cross have I fully checked for traffic; and if I have am I behind the traffic when turning cross and if not am I at least turning at a standard altitude of 500 feet above take off. And even still I need to be constantly scanning for traffic. Dipping my wings slightly could have helped me better spot the aircraft on my turn to downwind as that was my blind spot. As I am a new pilot I am not entirely familiar with process around an incident like this. It felt serious enough to me to warrant this submission. However Tower never indicated a severe issue; never gave me a number and merely told me 'frequency change approved' after a period of time had passed. That being said I would like to take responsibility for the mistakes that led to this deviation. At the very least writing it out helps me better understand and put to words the mistakes I made and how to better address them in the future.