Re: Skin Bridge
It's not totally clear what you're describing. You said it's a skin bridge at the "bottom" of your penis, but you meant the "bottom of the head (glans)," right? This can be a bit different depending on whether it was a result of circumcision (and most commonly it was).
But first, to answer your main question about sex with your girlfriend. It's possible it might hurt. If it hurts to have an erection at all, it will hurt even more to have sex than it does to masturbate (because in that case, you will likely have been masturbating very gently--more gently than sex usually is). IF present, the painful erections and pain from intercourse may go away as the skin stretches out to accommodate, though this longer skin may also be more noticeable. (You can basically see for yourself if it's going to hurt by tugging up and down on the shaft. If the tightness of the skin makes that painful, it will probably be the same for sex.)
If you were circumcised, the skin bridge you described is a fairly common circumcision complication (15% of the circumcised men seeing a urologist for any reason in one small study from 1984). The skin bridge is the result of inappropriate skin fusing during healing from the trauma of the circumcision.* Because of the strength of the scar tissue fusion that can form from trauma, it won't necessarily properly separate just from gentle tugging. (Of course strong tugging could damage your penis, so don't overdo it.) If the skin bridge causes other problems, you may need a (minor) surgery to correct it. The surgery will be expensive if you are uninsured, but otherwise your insurance should cover it. And there are good reasons to correct it: if it's uncomfortable, causes painful erections or crookedness. Also the skin bridges tend to grow longer in time making them more noticeable.
About skin bridges generally, there seem to be men who live with them and aren't bothered by having them. They've had them since they formed their body identity, so it feels perfectly normal. So I don't think you should change yourself just for what others consider normal--only if it's what you want. I think that even just the fear of surgery could be considered a valid reason to forgo it when considering the fact that there are always some risks in surgery and even a small mistake or complication could have sexual consequences. At the very least, it seems you should be sure your doctor is extremely well qualified to perform any surgery you get. On the other hand, if you would say you have any excessive or unpleasant emotional issues about it, that might be a good reason to have it corrected. The sort of negative mental programming you could give yourself over it (IF you were inclined to do so), would not be worth the risks of surgery (especially such a minor surgery).
But this is just my humble opinion. So do whatever you want to do about it. It's your body (not your parents' or anyone else's). I hope the info helped.
* Circumcision creates a fresh raw wound. Circumcision is usually performed in infancy when the normal preputial adhesion is still present, so one part of circumcision is roughly scrapping the adhering tissues apart.
If you are underage and uncomfortable talking to your parents about wanting corrective surgery, you could note that it's actually their fault for having you circumcised and then failing to tend to the wound properly afterward. The raw wound should have been checked occasionally while healing to keep it from fusing as it did (especially 10 to 14 days after the surgery). That was a parental obligation that their doctor should have informed them to do. I think a lot of men would agree that they "owe it to you" to get you the corrective surgery (if you want it). Of course it's likely they weren't adequately informed of this aftercare concern making them innocent victims of medical misinformation (inadequate information of potential risks, complications or even any effect of the procedure). Even the AAP task force that released the recent 'benefits outweigh risks' statement said directly, "We did not consider the risks."
You didn't mention foreskin, but if you are in fact intact/uncircumcised, there are a couple possibilities. The "skin bridge" may actually instead be your natural preputial adhesion. These are practically universal in infancy but become less common with age. (By puberty they are uncommon.) Such adhesions are easily corrected by regular gentle tugging. (See the thread "Stretching To Fix Common Foreskin Problems" under Men's Health--not Men's Sexual Health.) A natural adhesion can look like continuous/connected skin, but eventually it separates cleanly (with only a small amount of soreness afterwards). The other option (with foreskin) is that a tougher skin bridge can have resulted from premature/forcible foreskin retraction. This can have resulted in scarring (like that incurred from circumcision) making it more difficult/impossible to correct with just stretching. A doctor should be able to recognize if this is your problem.
Unfortunately American medical training has traditionally been so inadequate about foreskin that a very large number of doctors and nurses are very ignorant about basic development of the intact penis, hygiene, etc. This is why they perform premature/forcible foreskin retractions. This large number of doctors and especially nurses mistakenly believe it's good hygiene to retract the foreskin to clean infants' penises. They seem not to know that it's unnecessary to clean the inside of the genitals (until the child is old enough to do so for themselves).
Sorry about all the ancillary info. I wanted to explain those things about normal penis development and care.