I am writing a book a about a normal girl investigating her friend's murder. I wanted to ask if it's possible that a normal person who is not a detective could actually solve a murder?
Yes. Many cold cases (cases that remain unsolved and all leads provided by evidence have dried up) are kept alive by interested members of the public who want to find some closure to the victims, even if evidence does not exist to conclusively prove in trial or the criminal is deceased. Netflix has a good documentary series called "The Keepers" is about a group of women who are trying to solve a mystery surrounding the Murder of a beloved teacher and Nun at their Catholic High School in Baltimore and the group got together after accusations of sexual abuse by a priest at the same school and their own subsequent reveal of being victims (thinking they were alone). Most, if not all, former student investigators were all victims of the priest and through their shared experiences, have a working theory that the nun was murdered because one of the victims confided in her shortly before her murder (the priest in question died before he could be brought on any criminal charges).
Additionally, in the United States (and other common law nations) a civilian is allowed to detain someone they suspect is comitting a crime until such time that the police can to take the detained into custody (known as Citizen's arrest). Typically, this is if the criminal is seen in the act, but it can occur if the person is holding them against their will under enough evidence for the cops to make an arrest while the cops take their time arriving. Additionally, in the United States, the legal concept of "Castle Doctrine" allows for self defense against illegal intruders on your property even in States where "Stand your ground laws" do not exist for self defense on public property and extends to anyone who is invited to be on your property so long as the intruder is known to not be invited (So Kevin from Home Alone was well within his rights to set traps for the Wet Bandits, and many doctors have agreed that by the time both intruders were in the house, the injuries sustained would have prevented them from getting up to the first floor, if not outright kill them). At no point in the film is Kevin implausibly aware of facts that lead him to conclude the Bandits are up to no good (even though McCullcan seemed to have a talent for playing kids who were capable of behaving way more mature than a kid his age, nothing with respect to the bandits is overly out of the bounds of plausible for a 6-8 year old (I forget his age, it's been the better part of a year since I last watched it).
Yes, a citizen can investigate a murder. They don't have the tools of the police or courts, for example they can't force a store to release security tapes, or reveal employee records.
But if they do find evidence (and can prove it isn't fake) they can bring that to the police and possibly it will be useful in the apprehension or trial of that individual.
For example, they might trace the movements of the deceased, and with some effort and guesswork informed by their relationship with the deceased, find their path, and develop new suspects.
Private investigators usually have to be licensed (five US States do not require a license), but they are not police, cannot make anything but a Citizen's Arrest, and cannot compel somebody to produce records, they can't get a search warrant from a judge, etc.
But they can still ask people questions and investigate deaths, both murder and accidental. So can a normal person, you still have freedom of speech, freedom to look through public records, make connections.
I know from personal experience (two murders of people I know, separate incidents) that all those Law Enforcement shows are just fiction, most murders get very short shrift, without a witness the majority of murders go unsolved, forensics is sparse, police detectives hit a dead end and (if the case is not high profile) they move on to the next one, because there is an endless supply. There are clues that can be interpreted.
If you are talking about a fictional character, it would help if they have money. It is not illegal to pay somebody to talk to you. You can't pay them to be a witness, obviously, but it is my understanding (and I am not a lawyer) that you also cannot violate the privacy of a dead person.