The game is not designed to support your style of play. You cannot defeat mid-game challenges without fundamentally changing the mechanical basis of your character or the game. However, if you take a different class and call yourself a fighter without magic items, your narrative concept is preserved without ignoring the mechanical realities of the game.
The fundamental problem exists with the social contract you have with the rest of the group. While I applaud your determination to avoid "magic," the system and the mathematics presented by the system require said "magic."
Setting the fundamental problem aside, though. You present an interesting technical problem. You have, functionally, sworn a vow of poverty (in the book of exalted deeds) sense as a player, instead of a character. Furthermore, suggesting that you run a different class, one more able to be talented in the absence of magic is also basically out.
The short version is to give your PC the feat vow of poverty for free. Refluff it to support your concept of "mundane guy fighting dragons." The feat has problems, but the way you're playing your character, it should provide the necessary math-fixes to your character at the appropriate levels to simulate magical items. Another option is the leadership feat. you can substitute "quality" (magic items) with "quantity." Having a small army at your back can equalize many problems.
The longer version is to realize that a mundane guy... will indeed die when fighting dragons. If you choose not to refluff any existing options then you will indeed have harder and harder going. While this presents a very compelling campaign of a bunch of normal people trying to do a "hero's" job, that is something that everyone in the party will have to get behind. If you're a fighter in a group of magic-using and wielding people, and refuse to alter the mechanical basis of your character (not necessary the fluff of it), then the game will indeed be far more difficult for you than was otherwise intended.
What I would do, if I was playing a "mundane" character out to save the world is base it either on a factotum or one of the classes (probably warblade) from book of nine swords. Both offer acceptable in-game mechanical capabilities, even though they'll be horribly outclassed by the "magic" using people. Warblade is somewhat compatible with vow of poverty (especially one that allows you metal armor and weapons). The skill capabilities of the factotum (and if we refluff the light "magic" capabilities to be martial practices) also suit what I think your "find ways to cause the environment to kill the problems rather than me being some idiot in a tin suit running up to the dragon" methodology.
Therefore, my concept would be well served, but mechanically I would be able to play the same game as the rest of the party, supporting them in their fun without ruining my fun.
For more reading, see:
- Tier system for classes, this looks at the mechanical ability (instead of your preferred inventive ability) of classes to operate within and outside their areas of expertise. Huge variation in tier-capability can make people feel less powerful. Those feelings, unfortunately, are often correct.
- Constrained Optimization in D&D, My paper on optimization. While it looks at 4e in detail, it should give you some idea of the interactions between the mechanical-theoretical level and the story-narrative level. On the other hand, it may be particularly infuriating to you, as I don't privilege character-action over mechanics in any way. If you are likely to become offended at this sort of statistical treatment of game, I don't recommend reading it.