For any primorial $p_k \ge 3$, $p_k\#$, there are $$\prod_{2\le{i}\le{k}} (p_i-2)$$ distinct instances of $x,x+2$ that are relatively prime to $p_k\#$.
If any of these pairs are less than $p_{k+1}^2$, then they are necessarily twin primes.
For the heck of it, I wrote a tiny app that checks all the primes up to 191,137 and in each case there was at least one twin prime between $p_k^2$ and $p_{k+1}^2$.
Can it be proven that this eventually fails? Are there two consecutive primes $p_m, p_{m+1}$ such that if $p_i$ is a prime and $p_m^2 < p_i < p_{m+1}^2$, then $p_i,p_{i+1}$ are not a twin primes.
You're looking for a small interval, so suppose you have twin primes ($p_k+2=p_{k+1}$). Then the gap is roughly $4\sqrt{x}$ with numbers around $x$. Heuristics suggest that, on average, such an interval would contain about $$ \frac{8C_2\sqrt{x}}{\log^2 x} $$ twin primes, where $C_2\approx0.6601618158$ is the twin prime constant. If we treat the primes as being Poisson distributed, the chance that no primes would be found in the interval is $$ \exp\left(-\frac{8C_2\sqrt{x}}{\log^2 x}\right) $$
For example, if $x=191137^2$ (using your number) then the chance is about 1 in $10^{741}$. The probabilities drop off rapidly from there, so probably you can find a pair of twin primes in any such interval.