It's not normal and expected, period. There's only one way it's acceptable to fvck someone else while in a relationship. That's if you agree with your girlfriend to have an open relationship, if she doesn't accept you have three choices. You can go deceive her and do it anyways (will end tragically), you can dump her and find a girlfriend who will accept it or you can accept not fvcking someone else.
I think the main point to making this work is to not do it if it will hurt her feelings, which it obviously will not if she accepts an open relationship. Another point to make is fairness, if you can have the freedom to fvck someone else so should she.
An open relationship is one where you're in a relationship and have mutually agreed it's ok to fvck someone else, mostly without letting it affect the relationship. So clearly this approach requires you to be pretty strong emotionally, to not become attached to the others you fvck. It's also a good idea to inform the others you fvck that you're in an open relationship and you're doing it just for the sex.
This might seem like a lot to handle, and it is. I'm telling you because it will keep you out of trouble and you'll have to stick to it to not make a mess. It's perfectly possible to do this, but since it's not normal you must communicate it clearly.
The other alternative is to be a pvssy and not tell the truth to the person you supposedly love. That's what we commonly refer to as cheating, it only works as long as you're not caught. Eventually it will break down and she will know the truth, from that point you're fvcked. Not just because of the cheating itself, but because you broke the number one rule of healthy relationships which is trust and honesty.
Please use your brain before going near this.