This may be interpreting... but I get the sense reading your question that you are not only wondering what's normal, but also that you might like to have more friends, but haven't figured out how.
I'm going to try to touch on both thoughts...
What's the norm for work friendships?
There is no normal. As @brunoguardia says - the nature of the other folks in your work and your relative similarities/differences are a big part of it. And so is the nature of whether the other folks have time outside of work to socialize. One way that I baseline it is that some work groups as a whole are more social than others. If your group isn't all that socially active with each other outside of work, then it's not you, it's them.
In my mind, I usually have 3 categories:
Friendly colleagues - people that I have a healthy and positive work relationship with - we've helped each other, we know and respect each other. I have a reasonable belief that I could ask these folks for a work-related favor that was a bit outside their job description, and they'd help me out. I feel that way, because I'd do the same for them. But we're colleagues - we don't talk about our personal lives, we don't rely on each other for anything personal/emotional.
Friends at work - folks that I'm friends with. At work. But not so much outside of work. We can share a good joke at lunch. We may even talk about our personal lives and give each other some advice. But the friendship doesn't really evolve outside of work. Maybe we just don't have time to meet up in the evening, or maybe we are such different folks that we couldn't find much that we'd like in common.
Friends that have transcended work - I would be surprised if ANYONE had a ton of these people. These are folks who are close friends. I've been to their house, they've been to mine. We have had numerous meals together and supported each other. It's hard to sustain a great many of these friendships, they take care and feeding.
I see it somewhat as a pyramid - I think a lot of folks have friendly colleagues but not everyone gets lucky enough enough to find their closest friendships at work. I certainly didn't have more than maybe 1-2 such friendships until I got really lucky 15 years into my career and happened to find a lot of people that turned out to be this type of friend.
I don't normally expect work friends or professional colleagues to stay in contact past leaving a job. Maybe we get lucky - but even if we exchange personal contact info, my expectations are low. This is somewhat culturally dependent. I know that a friend & colleague of mine is Russian and tells me that she has kept in contact with friends through all phases of life -- and she kind of pities my North American self for not having such durable friendships!
How do I build friendships at work?
Being around when people have slack-time helps. Even if you don't have much to say - people DO appreciate an active listener. For a connection to move from "friendly" to "friends" someone has to bridge the "we only talk about work efforts" boundary. It doesn't have to be a profound leap - I've bonded over broken umbrellas, favorite place to order takeout, commuting challenges, etc. In fact, the superficial can often feel a lot safer for both of you.
Tips to start it off:
- Do hang out in the common areas. Even if it feels awkward. If you really, really, really feel awkward, bring a book. Something fun. Many people will be interested in something you are reading, and it's more clear that you are reading something fun if it's an actual book, and not a Kindle, etc. Be prepared to be distracted. That's the goal.
- When there's slack time - like lunch or when a meeting is delayed, or folks are just hanging around... ask questions. People love to talk about themselves. "What are you doing this weekend?" is good.
- Go ahead and make the first proposal - ask some folks if they want to get lunch out somewhere. You have a great excuse - you are the intern, you're here for only a short time, and you're not from the area -- where is a good place to eat out? Would anyone like to go with you?
- If there are folks you are friendly with that are leaving your group or the company - go ahead and ask for contact info to keep in touch. It's nice to be asked, and the worst they will say is "no".
Caveat - be the judge of how much slack time makes sense. Work is still important, and if no one else is hanging out then it's probably not a good use of time.
Cut yourself Slack....
Even though I'm saying make overtures, also cut yourself some slack. Sometimes groups just aren't friends. I remember that on my first internship, I was a Bachelor's candidate, working in a laboratory where everyone else was a Master's graduate with 10+ years of experience. The educational difference wasn't the big difference - it was age. I was 20 years old, and the next youngest was ~30 with at least 2 kids. Most folks were parents of kids. While I wanted to hang out after work (or just eat lunch with someone) - most of them were coming in early, working through lunch, and then going home to their families ASAP. And even the other interns in the office were 5+ years older than me - because it was really rare for the company to hire people who weren't Master's candidates. And in fact the other interns were part of a well-organized program, while I was a contract hire.
So ... I was lonely. I eventually found friends through activities I liked socially. I played music with a group that became friends through after-rehearsal ice cream. And I liked night clubs and made friends over after-club pizza. My diet was terrible, but I made some good friends. ;)
The work group liked me - I was hired for a 2nd year, and they recommended me for a great position as a permanent hire (which I took), and were incredibly supportive professionally. We just never hit it off as "friends".