It's been less than a century since the widespread use of antibotics started, and already we're seeing bacteria that have evolved immunities to the antibotics we use.
On the other hand, we've been using immune systems to to fight bacteria for millions of years, and bacteria evolve much faster than humans do. Why have bacteria not evolved immunities that let them completely overwhelm our immune systems and kill us all?
First of, not all infections are mediated by bacteria and not all bacteria are infectious. Also, not all parasitic bacteria lead to a strong infection or important health problem that cause any non-negligible selection pressure. Following the implicit logic of your post I will talk about infectious bacteria which cause "noticeable harm".
The mistake in your thinking is in the sentence
bacteria evolve much faster than humans do
While it is true that the mutation rate of base pair is higher and that the generation time is lower, the statement bacteria evolve much faster than humans do is made way too general and lead to your mistake here.
Infectious bacteria are selected to deal with the host immune system and the host is also selected to deal with the the parasite. Both virulence and penetrance evolve in this evolutionary arms race.
The evolution of host-parasite interaction is a very large field of study and it would take several books (and several books have been written on the subject; see this Amazon search) to summarize it. For some theoretical insights into the evolution of host and parasites, you might be interested in the book by Martin Nowak Evolutionary Dynamics: Exploring the equations of life
You will also note that a parasite does not have any "direct interest" in killing (or even harming) its host (thanks @jamesqf for his comment). You might want to have a look at the post Why do parasites sometimes kill their hosts?
I'd like to add to @Remi.b's excellent answer with a few additional points
To quote Lewis Thomas in Germs
Disease usually results from inconclusive negotiations for symbiosis, an overstepping of the line by one side, a biologic misinterpretation of borders.
The human host is a complex ecosystem. Many "pathogens" live quite happily in one compartment, having co-evolved to the point where detection and attack by immune system mediators doesn't happen (the microbe avoids it) and is unnecessary (attacking the microbe provides no benefit to the host). For example, enterobacteriaceae have a capsule that allows them to avoid binding of IgA in the gut (see Murray Medical Microbiology, Chapter 30). Disease occurs when that otherwise commensal microbe is introduced to a compartment where it doesn't belong. Here, as it often is, disease may not be beneficial for either the host or the pathogen.
Two other small points to consider:
Antibacterial resistance is not new. The spread is new. In fact, resistance genes have been around for at least tens of thousands of years (see my answer to another question)
Again, a (human) host is a complex ecosystem. It is not an arms race between two species. A successful microbe must negotiate for space and resources with many other species, and may, in fact, be a host itself.
Every pathogen is resistant to immune responses. That's why they're pathogens. "Normal" bacteria, that don't have resistance adaptations, are immediately destroyed by immune responses; that's why we're not immediately infected by the vast cloud of microbes that surround us.
Individuals with weakened immunity (such as AIDS patients) are infected by microbes that normally aren't considered pathogenic. Such microbes don't have most of the immune-resistance armory that true pathogens or even opportunists have; but even they have some immune resistance capacity, as proven by the fact that even AIDS patients aren't infected with, say, plant viruses or the billions of soil bacteria that surround us, but that lack immune resistance altogether.
Although all pathogens and opportunists have immune resistance, few if any have complete immune resistance. Why not? Because the selective pressure on hosts to develop and maintain effective immunity is extremely high. Those hosts that failed to protect against some form of infection are probably extinct.
Immune-system resistance is especially difficult due to how immunity works.
First, I make 2 assumptions in my explanation:
When a pathogen is present, the immune system samples pieces of it (epitopes) and works on "randomly" putting together antibodies that bind to that epitope. Once a match is found, the qualifying antibody proliferates and is deployed in full force. Each copy of the matching antibody binds to the offending pathogen on contact with good probability - think of it as a microscopic football tackle. Cell-like pathogens (bacteria, fungi, and protozoa) thoroughly covered in antibodies can't consume nutrients, attack cells, and in extreme cases they can't excrete toxins because exocytosis proteins are blocked in antibodies. Viruses covered in antibodies have their surface structures blocked, thoroughly preventing binding to host cells.
The only reason why antibiotics are "easy" to gain resistance to by comparison is that they don't change on their own. The ones most vulnerable to resistance rely on tricking bacteria into eating them (endocytosis), which quickly selects for bacteria that work around the vulnerable protein channel. In other cases, bacteria develop something that inactivates antibiotics.
There has recently been work on antibiotics that force their way into bacterial cells, effectively punching large holes in the membrane. These would be difficult to develop resistance to, especially if they operate via electrical potential or grab onto a wide variety of bacterial epitopes.
Additionally, bacteria would be in a lot of trouble if humans developed antibiotic-generator implants that work as a secondary immune system. Consider a technology that scans for pathogenic bacteria, builds computerized 3D models of them, and then uses a biochemical 3D printer chip to construct custom antibiotics after preliminary safety checks to verify that the custom antibiotic in question wouldn't have dangerous side effects.
