Is this platform game level winnable?
Jelly, 23 22 bytes

ŒṪạþ`§&lt;4NA1¦A»/T$¦ÐLṪṀ


Try it online!

New version, which now efficiently handles moves backwards. Takes input as a boolean matrix (row major). Returns 1 for winnable and 0 for not. 

Thanks to @AZTECCO for first suggesting using the indices of platforms, and @tsh for a test case that made me rethink my approach. 

Explanation

ŒṪ                     | Multidimensional truthy indices of each
  ạþ`                  | Outer table using absolute difference and with the same argument on both sides:
     §                 | Sum innermost lists
      &lt;4               | - Less than 4
        N              | Negate
         A1¦           | Absolute top row
                  ÐL   | Apply the following until no changes:
            A   $¦     | Absolute the rows indicated by the following:
             »/        | - Max of each column
               T       | - Truthy indices (i.e. where there’s a 1 in the column)
                    Ṫ  | Finally take the last row
                     Ṁ | Max


Step b step

(each matrix shown collapsed as a grid, with - = -1):

1. Input

0010
0001
1000
0010
0001
1000


2.ŒṪ

Multidimensional indices.

[[1, 3], [2, 4], [3, 1], [4, 3], [5, 4], [6, 1]]


3. ạþ`§&lt;4N

Convert to initial outer table with -1 for possible moves between columns and 0 where no move possible.

--0-00
--0--0
00--0-
-----0
0-0--0
00-00-


4. A1¦A»/T$¦ÐL

Absolute first row, then iteratively take each column with at least one 1 in it, and absolute the rows at those indices; results after each iteration shown.

Absolute row 1 -&gt;

110100
--0--0
00--0-
-----0
0-0--0
00-00-

-&gt; Absolute rows 1, 2, 4 -&gt;

110100
110110
00--0-
111110
0-0--0
00-00-

-&gt; Absolute rows 1, 2, 3, 4, 5 -&gt;

110100
110110
001101
111110
010110
00-00-

-&gt; Absolute rows 1, 2, 3, 4, 5, 6 -&gt;

110100
110110
001101
111110
010110
001001


5. ṪṀ

Finally, take the max of the last row

1


My previous 61 byte version has explanation in the edit history; thanks to @JonathanAllan for golfing 3 bytes off that one!