The separation of the interference pattern fringes does not depend on the width of the slits.
Each slit produces its own diffraction pattern.
As the slit gets wider the diffraction pattern gets narrower.
Where the two diffraction patterns overlap you get the interference pattern.
When the slits are narrow the diffraction spreads the light out and there is considerable overlap of the diffraction patterns and hence there are many fringes visible.
With wide slits and narrow diffraction patterns the light does not spread out as much and so the amount of overlap of the diffraction patterns is less so there are fewer fringes visible.
The diagrams from the HyperPhysics website are excellent and I have copied two of them to illustrate the effect of changing the slit width.
The diagram on the left shows the diffraction pattern due to a single slit.
The width of that pattern is controlled by the slit width.
In the double slit arrangement the diffraction patterns from each slit overlap and an interference pattern is produced whose intensity is modulated the by diffraction pattern due to a single slit.
The separation of the interference fringes depends on the slit separation.
If now the slit width is increased the width of the diffraction envelope is decreased whilst the separation of the interference fringes stays the same.
So less interference fringes will be visible.
Below the two diagrams there are a pair of photographs from the University of New South Wales website which illustrate the effect quite well and all you need to do is imagine what you would see if the diffraction envelope decreased in width.