What type of automaton is Google's Turing Doodle?
Assuming that:


we can add an arbitrarily large number of rows ("status lines")
the rows can be arbitrarily long
the tape is infinite


I tried to build an Alan Turing's Doodle machine configuration that emulates the machine $M_4$ described in "Small Turing Machines and generalized busy beaver competition" that has a halting problem that depends on an open (up to my knowledge) Collatz-like problem. The full picture is available here.



... so even if the A.T.'s Doodle is perhaps not Turing complete (due to the non-overlapping left-only jump operator available only in the first row), it is powerful enough to walk the fine line of (un)decidability :-D

EDIT: TURING DOODLE IS TURING COMPLETE

(I leave the previous answer above, because I'm not sure that this part is correct :-)

I think that even with a single left non-overlapping jump the Turing Doodle is Turing complete!. The (simple) idea is to use the tape itself to store the current state and use multiple cells to represent a larger alphabet.

For example a 2 states 8 symbols TM can be simulated using the following tape representation:

    HEAD POSITION
    v
...[s][b2 b1 b0] [_][b2 b1 b0] [_][b2 b1 b0] ....
   ^^^^^^^^^^^^^
    "macro cell"


The Turing doodle can:


read $s$ and branch to one of the two different horizontal areas representing STATE A and STATE B;
read $b2, b1, b0$ and branch to a row representing one of the eight symbols;
write the next symbol, move the head to the "macro cell" on the left or right, and store on it the next state; in the figure below these operations (that can be done on a sequence of cells using the actions move left/right and write) are called "MW";
finally transfer the control to the upper row that with a single left jump will bring the control back to step 1.


The full picture is available here.



In the same manner, given a $TM$ with an arbitrary number of states and alphabet symbols we can build an equivalent Turing Doodle machine $DM$.