Can the turing machine accept these type of languages in O(n)? or better? $L = \{a^n b^n c^n ... d^n | n > 0\}$
Yes a TM with multiple head for example can easily do this. for $a^nb^n$ you can have two heads one at the first occurence of A's and other at the first occurence of B. Then at each iteration you can cross each off. If both see blank accept. If A head does not see A or B head does not see a B reject. Else cross both off with some special character.

Clearly this takes one linear pass in time O(N). Now for the lower bound. It is not possible to do with less than O(N). If you use N number of tapes by definition then it is $\Omega{(N)}$. If you use K number of tapes then it is $K \times (N/K)$ which is still O(N). The reason for this lower bound is that you have check each character since any one character can break the rule of this language. Thus we get a linear lower bound.