GA pilot reported a NMAC with previously departed aircraft due to their different performance characteristics. Pilot was hesitant to accept clearance but concern for landing traffic compelled him to accept takeoff clearance.

Narrative

Departing IFR on personal flight with commercial/CFI rated passenger (not receiving instruction). Departing IFR on ANAHEIM 1 departure from 26L was told to line up and wait behind Aircraft Y. Almost immediately after Aircraft Y lifted off I was cleared for immediate takeoff and told there was traffic on short final. I felt uncomfortable with the spacing but worried that with traffic on short final if I delayed it would cause a conflict since aircraft behind me would have to go around and 30 as well as 26R were being used as well.On takeoff and initial climb I noticed after raising gear and flaps that we were rapidly approaching Aircraft Y still climbing straight out but were still below 800' which is altitude specified on SID for the turn. Due to concern for imminent collision I made an evasive early right turn. While in the turn tower called out the traffic and authorized early turn. I reported traffic in sight and remainder of flight was uneventful.I debriefed the incident with the passenger the following day and felt that the spacing provided was not adequate for the two different aircraft types with drastically different performance profiles. A Bonanza and Cub (for example) have very different performance profiles and it seems it would be prudent to take this into account when considering spacing for departures; especially when both aircraft have significant initial segment on same heading.