Comparing $\mathbf{NP}$ and $\mathbf{E}$
This was a comment, but it became too long. I want to give an explicit example where the input size increases. Take the classic reduction from CircuitSAT to 3SAT. The usual idea is to assign a variable to every gate in the circuit. The variable's value is the output of the gate. The you add the constraints that make a gate's output reflect the gate's input and gate type. So if a gate (whose variable is g_1) is an AND gate and has input wires coming from gates g_2 and g_3, you'll get a constraint like g_1 = g_2 AND g_3. We also have variables for the inputs, of course.

With this reduction, even though there are n inputs, if the size of the circuit is, say, $O(n^5)$, then the resulting SAT instance will have $O(n^5)$ variables, which will require $O(2^{n^5})$ time to solve by brute forcing on a deterministic TM.

You also asked why E is not closed under Karp reductions. That's easy. Take a problem that only be solved in $O(2^{n^2})$ time, and therefore it is not in E.  Now we can Karp-reduce this to a language in E by padding the input with $O(n^2)$ zeros. The language in E to which we're reducing is the original language with each input of size n padded with $O(n^2)$ zeros. Now since the input size is $O(n^2)$ and we know an algorithm that solves this in $O(2^{n^2})$ time, this problem is in E.  So we reduced a problem not in E to a problem in E, showing that E is not closed under Karp reductions.