Turning over is not allowed.
In the original Tsuro game the tiles are square and have two points on each side, one third and two thirds of the way along the side and arcs connecting these eight points in various ways.
The shapes of the arcs aren't significant, only which two points they connect is.
There are 35 tiles, agreeing with the entry a(4) = 35 here.
If we vary the shape of the tile and the number of points per side (pps), we get the following table.
....pps:..0....1......2......3......4......5......6......7......8......9.....10

circle....1....0......1......0......2......0......5......0.....18......0....105 (A007769)
monogon...1....0......1......0......3......0.....15......0....105......0....945 (A001145)
digon.....1....1......3.....11.....65....513...5363..68219 .................... (A132101)
triangle..1....0......7......0...3483......0.............0.............0
square....1....2.....35...2688.508277 ......................................... (this entry)
pentagon..1....0....193......0.............0.............0.............0
hexagon...1....5...1799
heptagon..1....0..19311......0.............0.............0.............0
octagon...1...18.254143
9gon.....1....0.............0.............0.............0.............0
10gon....1..105
The pps = 2 column is A132102. Blank entries all represent numbers greater than one million.
A monogon is distinct from a circle in that a monogon has not just one side, but also one vertex. Monogons and digons can't exist with straight sides, of course, at least not on a flat plane, but there's no rule that says these tiles have to have straight sides.
If we allow reflections the numbers are smaller (this would be appropriate for a game where the tiles were transparent and could be flipped over):
....pps:..0....1......2......3......4......5......6......7......8......9.....10

circle....1....0......1......0......2......0......5......0.....17......0.....79 (A054499)
monogon...1....0......1......0......3......0.....11......0.....65......0....513 (A132101)
digon.....1....1......3......8.....45....283...2847..34518.511209 ............. (A132103)
triangle..1....0......7......0...1907......0.............0.............0
square....1....2.....30...1447.257107 ......................................... (A132104)
pentagon..1....0....137......0.............0.............0.............0
hexagon...1....5...1065
heptagon..1....0..10307......0.............0.............0.............0
octagon...1...17.130040
9gon.....1....0.............0.............0.............0.............0
10gon....1...79
The pps = 2 column is A132105.
