I was wondering if there existed a way to fix the difference in the running speeds of a PAL NES to run on TV's in the United States? Be it some converter or messing with the NES itself.
You'll have to get a PAL to NTSC converter.
A modification to make a PAL NES to output NTSC video would be extremely difficult because both the console and the games are designed for the video signal.
The NES's clock frequencies were chosen to match the timing of the video signal, so you'd have to replace the clock circuitry. The PPU is hard-wired to generate either NTSC or PAL video, so you'd have replace that too.
After that, games designed for NTSC consoles might work if you disabled the CIC, but PAL games won't run properly:
Games relying on precise timing (complex raster effects, for example) may have problems.
For some games, the shorter VBlank time on NTSC will cause major graphical glitches (scrolling will be messed up and tiles will written to the wrong places).
Games will run sped up due to being run at 60 Hz instead of 50.
Audio won't sound right because audio generation is based on the clock.
Rather than convert the NES, just get a PAL monitor. The common auto backup video systems use an NTSC/PAL switchable monitor, available for not much money.Backup monitor 7" It is also possible, with some video capture cards, to select PAL (then your computer display will allow you to use the game).
If your TV takes PAL input (some do: check all the menus), it's easier, of course.
I have a Dell 2007FP video monitor that takes PAL as well as NTSC, DVI, VGA...
The easiest and cheapest solution is probably a composite to HDMI adapter that supports PAL. But it will add about 2 frames of lag (search in that link for "HDV 8S" for the review), so fast-paced games may be a little more difficult to play.
For analog displays, the discussion gets a little more complicated.
You're never going to convert a 50 FPS to 60 FPS (or vice-versa) and have the game run perfectly...the way the designers meant it to run. It's always going to run too fast (PAL running on NTSC) or too slow with lag (NTSC running on PAL).
The problem is the conversion itself. Think of two metronomes starting at the same time. One runs at 50 FPS and the other at 60 FPS. Only once per 50*60=3000 ticks will they both be in sync. So most of the time, you will hear two clicks.
If you want the best experience, then get a monitor or TV that can lock onto both rates. Some TV's will sync to both NTSC and PAL.
Or, just get a PAL monitor...but that might be expensive having it shipped from Europe...unless you find someone who already has one near you.