How did I end up with this FizzBuzz?

C, 81 78 bytes

l,i;f(n){for(l=i=0;l<n;l+=++i%3?i%5?snprintf(0,0,"%d",i):4:i%5?4:8);return i;}

68 bytes if you don't mind converting to double and back:

l,i;f(n){for(l=i=0;l<n;l+=++i%3?i%5?log10(i)+1:4:i%5?4:8);return i;}

Jelly,  16  14 bytes

^{2 bytes saved using more recent language features ) for µ€ and Ä for +\}

3,5ḍS×4oDL$)Äi

Try it online! or see the test cases.

How?

Builds a list of the lengths of every item from 1 to the input, reduces by addition and then finds the one-based index of the input in the list. (This also means an invalid input results in 0, "not in list").

3,5ḍS×4oDL$)Äi - Main link: theLength
           )    - perform the chain to the left for each (€) in
                     implicit range from 1 to the input and
                     pass the result into the monadic chain (µ) to the right
3,5            - 3 paired with 5: [3,5]
   ḍ           - divides?  for a multiple of 15 [1,1]; sum = 2; times 4 = 8
    S          - sum       for a multiple of  5 [0,1]; sum = 1; times 4 = 4
     ×4        - times 4   for a multiple of  3 [1,0]; sum = 1; times 4 = 4
                           for none of those    [0,0]; sum = 0; times 4 = 0
          $    - last two links as a monad
        D      -     to decimal digit list
         L     -     length - e.g. 313 -> [3,1,3] -> 3
       o       - logical or: replace a 0 with the decimal length, keep the 4s and 8s
            Ä  - reduce with addition: e.g. [1,1,4,1, 4, 4, 1, 1, 4, 4, 2, 4, 2 ,2, 8]
                                         -> [1,2,6,7,11,15,16,17,21,25,27,31,33,35,43]
             i - index of theLength in that list (e.g. 15 is at index 6)

MATL, 31 28 27 bytes

`@:tI5h!\XJA)VXznJ~z4*+G-}@

Try it online!

Explanation

`        % Do...while
  @:     %   Push array [1 2 ...k], where k is iteration index
  t      %   Duplicate  
  I5h!   %   Push column vector [3; 5]
  \      %   Modulo, with broadcast. Gives 2 × k matrix
  XJ     %   Copy into clipboard J
  A      %   Row vector that contains true for columns that contain two nonzeros
  )      %   Index with that vector. This keeps numbers that are non-fizz/buzz
  V      %   Convert to string. This inserts spaces between numbers
  Xzn    %   Number of nonspace characters
  J      %   Push 2 × k matrix resulting from modulo operation again
  ~z     %   Number of zeros
  4*     %   Multiply by 4. Gives number of characters corresponding to fizz/buzz
  +      %   Add
  G-     %   Subtract input. This is the loop condition: exit if 0
}        % Finally (execute right before exiting loop)
  @      %   Push current iteration index
         % End (implicit)
         % Display (implicit)