I should say "problems". I think they're two-fold:
There's no such thing as "forensic science" - there's just science. The term "forensic" refers to how the science is used, not to the underlying scientific concepts or procedures. Instead, the term is treated as a brand - as something that sets the science apart from routine, day-to-day science (and, it seems, often exempts "forensic science" from the scientific method).
Forensic science is driven by police and prosecutors. Crime scene investigation is performed by police officers, who have attended a weeks-long police training course and perhaps some additional seminars. The evidence they collect is submitted to government funded "crime laboratories" that are sometimes deplorable. The credentialed professionals who may form the backbone of a prosecutor's case may turn out to be well-spoken incompetents.
Occasionally I read about hopes to shake up the system, bring in real science, and there have been modest improvements in some areas, such as fire science as applied by arson investigators. Absent a scandal there's little to no movement for reform from within the system. Yet we have these scandals on a cyclical basis, and promises of reform (or potential for reform) never seem to take root. What is probably the best solution, adequately and independently funding defense experts, isn't going to happen.