Well once you get better equipment, people will go down quicker (like raiders) and I guess that might help you get over the slowness. I know it was rough for me at first, I only started playing the originals like 4 months ago so I'm not exactly an expert on the game though.
Well, you could always try to imagine what the scene would look like in real-time when you play it. Like, what the combat represents rather than what its battlefield looks like. For example, when you are surrounded by 6 rats I like to imagine that my character is actually lying on the ground with the rats crawling all over the character, biting and clawing away at her while she tries to shoot or hit them one by one. Or when a raider shoots me and hits me I like to imagine that it only grazed my skin or that the bullet was but a fleshwound. Once you start to imagine how it would actually look like you can let your creativity flow free and it won't seem as ridiculous. Turn-based games have the tendency to look rather ridiculous, especially when you're standing up hitting rats with your fists. But if you think about how the scene would actually play out it is far more interesting.
At least that's what I do in turn-based games and it makes the battles far more fun and interesting as I'm filling in the gaps of what can't be represented.
Another thing could be that if you are just interested in playing them for their stories you could always use an editor to buff your character to be nearly invincible. That way you can easily go through the combat and just move on to the story parts. Though by doing this you won't get a proper understanding of what enemies are truly dangerous at your level and who aren't. Like, taking on Metzgers Slavers Guild right off the bat is crazy but if you level up some more and get some companions the battle is far more even. So by cheating with your character you won't get the right feel for the combat scenarios.